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Search: a326551 -id:a326551
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A326556
E.g.f. C(x)^2 = Sum_{n>=0} a(n)*x^(2*n)/(2*n)!^2, where C(x) = cos( Integral 1/x * (Integral C(x) dx) dx ) is the e.g.f of A326551.
+20
1
1, -4, 256, -67072, 49479680, -82817122304, 273099601739776, -1606512897507196928, 15659025634284911198208, -238894370882781809622384640, 5451274531297360096585324691456, -179296966081016547805899589056200704, 8242844472527700570663352676068232265728, -516102091343047279882754030489835708929277952, 43042816831864259208854418353099287467922680709120
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
COMMENTS
The e.g.f. C(x)^2 can be derived from the functions described by A326800, A326801, and A326802.
LINKS
Table of n, a(n) for n=0..14.
EXAMPLE
E.g.f.: C(x)^2 = 1 - 4*x^2/2!^2 + 256*x^4/4!^2 - 67072*x^6/6!^2 + 49479680*x^8/8!^2 - 82817122304*x^10/10!^2 + 273099601739776*x^12/12!^2 - 1606512897507196928*x^14/14!^2 + 15659025634284911198208*x^16/16!^2 - 238894370882781809622384640*x^18/18!^2 + 5451274531297360096585324691456*x^20/20!^2 + ...
where C(x) is the e.g.f. of A326551:
C(x) = 1 - 2*x^2/2!^2 + 56*x^4/4!^2 - 8336*x^6/6!^2 + 3985792*x^8/8!^2 - 4679517952*x^10/10!^2 + 11427218287616*x^12/12!^2 - 51793067942397952*x^14/14!^2 + 400951893341645930496*x^16/16!^2 - 4975999084909976839454720*x^18/18!^2 + 94178912073481319162642169856*x^20/20!^2 -+ ...
such that C(x) = cos( Integral 1/x * (Integral C(x) dx) dx ),
note also C(x*y) = cos( Integral Integral C(x*y) dx dy ).
PROG
(PARI)
{a(n) = my(C=1, S=x); for(i=1, 2*n,
S = intformal( C/x * intformal( C +x*O(x^(2*n)) ) );
C = 1 - intformal( S/x * intformal( C +x*O(x^(2*n)) ) ); ); (2*n)!^2*polcoeff(C^2, 2*n)}
for(n=0, 30, print1(a(n), ", "))
CROSSREFS
Cf. A326551, A326800, A326801, A326802.
KEYWORD
sign
AUTHOR
Paul D. Hanna, Jul 28 2019
STATUS
approved
A326552
E.g.f. S(x), where C(x*y) + iS(x*y) = exp( i*Integral Integral C(x*y) dx dy ) such that C(x)^2 + S(x)^2 = 1.
+10
8
1, -8, 576, -160768, 123535360, -212713734144, 716196297048064, -4280584942657732608, 42250703121584165486592, -651154631135458759089848320, 14983590319172065236171175755776, -496301942561421311900528265903734784, 22953613919171561374366988621726483480576, -1444609513446024762131466039751756562435145728, 121022534222796916421149671221445519229890299166720
(list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
The hyperbolic analog of the e.g.f. is described by A325292.
The e.g.f. can be derived from the functions described by A326797, A326798, and A326799.
The e.g.f. can be derived from the functions described by A326800, A326801, and A326802.
LINKS
Table of n, a(n) for n=1..15.
FORMULA
E.g.f. S(x) = Sum_{n>=0} a(n)*x^(2*n+1)/(2*n+1)!^2, where series S(x) and related series C(x) satisfy the following relations.
(1.a) C(x)^2 + S(x)^2 = 1.
(1.b) S'(x)/C(x) = -C'(x)/S(x) = 1/x * Integral C(x) dx.
(2.a) S(x) = Integral C(x)/x * (Integral C(x) dx) dx.
(2.b) C(x) = 1 - Integral S(x)/x * (Integral C(x) dx) dx.
(3.a) C(x) + i*S(x) = exp( i*Integral 1/x * (Integral C(x) dx) dx ).
(3.b) C(x) = cos( Integral 1/x * (Integral C(x) dx) dx ).
(3.c) S(x) = sin( Integral 1/x * (Integral C(x) dx) dx ).
Integration.
(4.a) S(x*y) = Integral C(x*y) * (Integral C(x*y) dy) dx.
(4.b) C(x*y) = 1 - Integral S(x*y) * (Integral C(x*y) dy) dx.
(4.c) S(x*y) = Integral C(x*y) * (Integral C(x*y) dx) dy.
(4.d) C(x*y) = 1 - Integral S(x*y) * (Integral C(x*y) dx) dy.
Exponential.
(5.a) C(x*y) + i*S(x*y) = exp( i*Integral Integral C(x*y) dx dy ).
(5.b) C(x*y) = cos( Integral Integral C(x*y) dx dy ).
(5.c) S(x*y) = sin( Integral Integral C(x*y) dx dy ).
Derivatives.
(6.a) d/dx S(x*y) = C(x*y) * Integral C(x*y) dy.
(6.b) d/dx C(x*y) = -S(x*y) * Integral C(x*y) dy.
(6.c) d/dy S(x*y) = C(x*y) * Integral C(x*y) dx.
(6.d) d/dy C(x*y) = -S(x*y) * Integral C(x*y) dx.
EXAMPLE
E.g.f. S(x) = x - 8*x^3/3!^2 + 576*x^5/5!^2 - 160768*x^7/7!^2 + 123535360*x^9/9!^2 - 212713734144*x^11/11!^2 + 716196297048064*x^13/13!^2 - 4280584942657732608*x^15/15!^2 + 42250703121584165486592*x^17/17!^2 - 651154631135458759089848320*x^19/19!^2 + 14983590319172065236171175755776*x^21/21!^2 -+ ...
where S(x) = sin( Integral 1/x * (Integral C(x) dx) dx ),
also, S(x*y) = sin( Integral Integral C(x*y) dx dy ),
such that C(x)^2 + S(x)^2 = 1.
RELATED SERIES.
C(x) = 1 - 2*x^2/2!^2 + 56*x^4/4!^2 - 8336*x^6/6!^2 + 3985792*x^8/8!^2 - 4679517952*x^10/10!^2 + 11427218287616*x^12/12!^2 - 51793067942397952*x^14/14!^2 + 400951893341645930496*x^16/16!^2 - 4975999084909976839454720*x^18/18!^2 + 94178912073481319162642169856*x^20/20!^2 -+ ...
where C(x) = cos( Integral 1/x * (Integral C(x) dx) dx ),
also, C(x*y) = cos( Integral Integral C(x*y) dx dy ).
RELATED FUNCTIONS.
Given functions Ax, Bx, Cx, Ay, By, and Cy defined by
(1a) Ax = 0 + Integral Bx*Cy - Cx*By dx,
(1b) Bx = 1 + Integral Cx*Ay - Ax*Cy dx,
(1c) Cx = 0 + Integral Ax*By - Bx*Ay dx,
(2a) Ay = 0 + Integral By*Cx - Cy*Bx dy,
(2b) By = 0 + Integral Cy*Ax - Ay*Cx dy,
(2c) Cy = 1 + Integral Ay*Bx - By*Ax dy,
then
S(x*y) = Ax*Ay + Bx*By + Cx*Cy.
These related series begin as follows.
Ax = x + (-1*x^3 - 3*x*y^2)/3! + (1*x^5 - 30*x^3*y^2 + 5*x*y^4)/5! + (-1*x^7 + 315*x^5*y^2 + 525*x^3*y^4 - 7*x*y^6)/7! + (1*x^9 - 1260*x^7*y^2 + 18270*x^5*y^4 - 2940*x^3*y^6 + 9*x*y^8)/9! + (-1*x^11 + 3465*x^9*y^2 - 496650*x^7*y^4 - 695310*x^5*y^6 + 10395*x^3*y^8 - 11*x*y^10)/11! + ... (A326797)
Bx = 1 + (-1*x^2)/2! + (1*x^4)/4! + (-1*x^6 + 120*x^4*y^2)/6! + (1*x^8 - 672*x^6*y^2)/8! + (-1*x^10 + 2160*x^8*y^2 - 120960*x^6*y^4)/10! + (1*x^12 - 5280*x^10*y^2 + 1584000*x^8*y^4)/12! + ... (A326798)
Cx = (2*x*y)/2! + (-4*x*y^3)/4! + (-160*x^3*y^3 + 6*x*y^5)/6! + (1344*x^3*y^5 - 8*x*y^7)/8! + (145152*x^5*y^5 - 5760*x^3*y^7 + 10*x*y^9)/10! + (-2534400*x^5*y^7 + 17600*x^3*y^9 - 12*x*y^11)/12! + ... (A326799)
Ay = -y + (3*x^2*y + 1*y^3)/3! + (-5*x^4*y + 30*x^2*y^3 + -1*y^5)/5! + (7*x^6*y + -525*x^4*y^3 + -315*x^2*y^5 + 1*y^7)/7! + (-9*x^8*y + 2940*x^6*y^3 + -18270*x^4*y^5 + 1260*x^2*y^7 + -1*y^9)/9! + (11*x^10*y + -10395*x^8*y^3 + 695310*x^6*y^5 + 496650*x^4*y^7 + -3465*x^2*y^9 + 1*y^11)/11! + ...
By = (2*x*y)/2! + (-4*x^3*y)/4! + (6*x^5*y + -160*x^3*y^3)/6! + (-8*x^7*y + 1344*x^5*y^3)/8! + (10*x^9*y + -5760*x^7*y^3 + 145152*x^5*y^5)/10! + (-12*x^11*y + 17600*x^9*y^3 + -2534400*x^7*y^5)/12! + ...
Cy = 1 + (-1*y^2)/2! + (1*y^4)/4! + (120*x^2*y^4 + -1*y^6)/6! + (-672*x^2*y^6 + 1*y^8)/8! + (-120960*x^4*y^6 + 2160*x^2*y^8 + -1*y^10)/10! + (1584000*x^4*y^8 + -5280*x^2*y^10 + 1*y^12)/12! + ...
PROG
(PARI) {a(n) = my(C=1, S=x); for(i=1, 2*n+1,
S = intformal( C/x * intformal( C +x*O(x^(2*n+1)) ) );
C = 1 - intformal( S/x * intformal( C +x*O(x^(2*n+1)) ) ); ); (2*n+1)!^2*polcoeff(S, 2*n+1)}
for(n=0, 30, print1(a(n), ", "))
CROSSREFS
Cf. A326551, A325292.
Cf. A326797, A326798, A326799.
Cf. A326800, A326801, A326802.
KEYWORD
sign
AUTHOR
Paul D. Hanna, Jul 25 2019
STATUS
approved
A326800
Consider the e.g.f. S(x,y) = Sum_{n>=0} Sum_{k=0..n} T(n,k) * x^(2*n-2*k+1) * y^(2*k) / ((2*n-2*k+1)!*(2*k)!) and related functions C(x,y) and D(x,y), as defined in the Formula section. Sequence gives the triangular array of coefficients T(n,k) (n>=0, 0<=k<=n) of S(x,y).
+10
8
1, -1, -1, 1, -3, 1, -1, 15, 15, -1, 1, -35, 145, -35, 1, -1, 63, -1505, -1505, 63, -1, 1, -99, 5985, -30387, 5985, -99, 1, -1, 143, -16401, 539679, 539679, -16401, 143, -1, 1, -195, 36465, -3275811, 18679617, -3275811, 36465, -195, 1, -1, 255, -70785, 12723711, -506849409, -506849409, 12723711, -70785, 255, -1
(list; table; graph; refs; listen; history; text; internal format)
OFFSET
1,5
COMMENTS
The e.g.f. S(x,y) is equivalent to the e.g.f. of A326797.
LINKS
Paul D. Hanna, Table of n, a(n) for n = 1..1891 (first 61 rows of this triangle).
FORMULA
The e.g.f. Sx = S(x,y) and related functions Cx = C(x,y), Dx = D(x,y), Sy = S(y,x), Cy = C(y,x), and Dy = D(y,x) satisfy the following relations.
DEFINITION.
(1a) Sx = Integral Cx*Dy + Cy*Dx dx,
(1b) Cx = sqrt(1/2) - Integral Sx*Dy + Sy*Dx dx,
(1c) Dx = sqrt(1/2) - Integral Sx*Cy - Sy*Cx dx,
(2a) Sy = Integral Cy*Dx + Cx*Dy dy,
(2b) Cy = sqrt(1/2) - Integral Sy*Dx + Sx*Dy dy,
(2c) Dy = sqrt(1/2) - Integral Sy*Cx - Sx*Cy dy.
IDENTITIES.
(3a) Dx^2 + Cx^2 + Sx^2 = 1.
(3b) Dy^2 + Cy^2 + Sy^2 = 1.
(4a) Dx*(d/dx Dx) + Cx*(d/dx Cx) + Sx*(d/dx Sx) = 0.
(4b) Dy*(d/dy Dy) + Cy*(d/dy Cy) + Sy*(d/dy Sy) = 0.
(4c) Dy*(d/dx Dx) - Cy*(d/dx Cx) - Sy*(d/dx Sx) = 0.
(4d) Dx*(d/dy Dy) - Cx*(d/dy Cy) - Sx*(d/dy Sy) = 0.
(5a) (Dx*Dy - Cx*Cy - Sx*Sy)^2 + (d/dx Dx)^2 + (d/dx Cx)^2 + (d/dx Sx)^2 = 1.
(5b) (Dx*Dy - Cx*Cy - Sx*Sy)^2 + (d/dy Dy)^2 + (d/dy Cy)^2 + (d/dy Sy)^2 = 1.
RELATED FUNCTIONS.
(6a) SS(x*y) = Dx*Dy - Cx*Cy - Sx*Sy.
(6b) d/dx SS(x*y) = Dx*(d/dx Dy) - Cx*(d/dx Cy) - Sx*(d/dx Sy).
(6c) d/dy SS(x*y) = Dy*(d/dy Dx) - Cy*(d/dy Cx) - Sy*(d/dy Sx).
(7a) CC(x*y)^2 = (Cx*Dy + Cy*Dx)^2 + (Sx*Dy + Sy*Dx)^2 + (Sx*Cy - Sy*Cx)^2.
(7b) CC(x*y)^2 = (d/dx Dx)^2 + (d/dx Cx)^2 + (d/dx Sx)^2.
(7c) CC(x*y)^2 = (d/dy Dy)^2 + (d/dy Cy)^2 + (d/dy Sy)^2.
In the above, CC(x) and SS(x) are the e.g.f.s of A326551 and A326552 defined by
(8a) CC(x*y)^2 + SS(x*y)^2 = 1,
(8b) SS(x*y) = Integral CC(x*y) * (Integral CC(x*y) dy) dx,
(8c) CC(x*y) = 1 - Integral SS(x*y) * (Integral CC(x*y) dy) dx,
(8d) SS(x*y) = sin( Integral Integral CC(x*y) dx dy ),
(8e) CC(x*y) = cos( Integral Integral CC(x*y) dx dy ).
DERIVATIVES.
(9a) d/dx Sx = Cx*Dy + Cy*Dx.
(9b) d/dx Cx = -Sx*Dy - Sy*Dx.
(9c) d/dx Dx = -Sx*Cy + Sy*Cx.
(9d) d/dy Sy = Sy*Dx + Sx*Dy.
(9e) d/dy Cy = -Sy*Dx - Sx*Dy.
(9f) d/dy Dy = -Sy*Cx + Sx*Cy.
EXAMPLE
E.g.f.: S(x,y) = x + (-x^3/3! - x*y^2/2! ) + ( x^5/5! - 3*x^3*y^2/(3!*2!) + x*y^4/4! ) + (-x^7/7! + 15*x^5*y^2/(5!*2!) + 15*x^3*y^4/(3!*4!) - x*y^6/6! ) + ( x^9/9! - 35*x^7*y^2/(7!*2!) + 145*x^5*y^4/(5!*4!) - 35*x^3*y^6/(3!*6!) + x*y^8/8! ) + (-x^11/11! + 63*x^9*y^2/(9!*2!) - 1505*x^7*y^4/(7!*4!) - 1505*x^5*y^6/(5!*6!) + 63*x^3*y^8/(3!*8!) - x*y^10/10! ) + ( x^13/13! - 99*x^11*y^2/(11!*2!) + 5985*x^9*y^4/(9!*4!) - 30387*x^7*y^6/(7!*6!) + 5985*x^5*y^8/(5!*8!) - 99*x^3*y^10/(3!*10!) + x*y^12/12! ) + (-x^15/15! + 143*x^13*y^2/(13!*2!) - 16401*x^11*y^4/(11!*4!) + 539679*x^9*y^6/(9!*6!) + 539679*x^7*y^8/(7!*8!) - 16401*x^5*y^10/(5!*10!) + 143*x^3*y^12/(3!*12!) - x*y^14/14! ) + ( x^17/17! - 195*x^15*y^2/(15!*2!) + 36465*x^13*y^4/(13!*4!) - 3275811*x^11*y^6/(11!*6!) + 18679617*x^9*y^8/(9!*8!) - 3275811*x^7*y^10/(7!*10!) + 36465*x^5*y^12/(5!*12!) - 195*x^3*y^14/(3!*14!) + x*y^16/16! ) + (-x^19/19! + 255*x^17*y^2/(17!*2!) - 70785*x^15*y^4/(15!*4!) + 12723711*x^13*y^6/(13!*6!) - 506849409*x^11*y^8/(11!*8!) - 506849409*x^9*y^10/(9!*10!) + 12723711*x^7*y^12/(7!*12!) - 70785*x^5*y^14/(5!*14!) + 255*x^3*y^16/(3!*16!) - x*y^18/18! ) + ( x^21/21! - 323*x^19*y^2/(19!*2!) + 124865*x^17*y^4/(17!*4!) - 38067315*x^15*y^6/(15!*6!) + 4363117473*x^13*y^8/(13!*8!) - 26803260803*x^11*y^10/(11!*10!) + 4363117473*x^9*y^12/(9!*12!) - 38067315*x^7*y^14/(7!*14!) + 124865*x^5*y^16/(5!*16!) - 323*x^3*y^18/(3!*18!) + x*y^20/20! ) + ...
This triangle of coefficients T(n,k) of x^(2*n-2*k+1)*y^(2*k)/((2*n-2*k+1)!*(2*k)!) in e.g.f. S(x,y) begins
1;
-1, -1;
1, -3, 1;
-1, 15, 15, -1;
1, -35, 145, -35, 1;
-1, 63, -1505, -1505, 63, -1;
1, -99, 5985, -30387, 5985, -99, 1;
-1, 143, -16401, 539679, 539679, -16401, 143, -1;
1, -195, 36465, -3275811, 18679617, -3275811, 36465, -195, 1;
-1, 255, -70785, 12723711, -506849409, -506849409, 12723711, -70785, 255, -1;
1, -323, 124865, -38067315, 4363117473, -26803260803, 4363117473, -38067315, 124865, -323, 1;
-1, 399, -205105, 95686591, -22813329825, 1031421316783, 1031421316783, -22813329825, 95686591, -205105, 399, -1;
1, -483, 318801, -212188067, 88405315713, -11952302851203, 77353020714385, -11952302851203, 88405315713, -212188067, 318801, -483, 1; ...
RELATED SERIES.
The e.g.f. of A326801 begins
C(x,y) = sqrt(1/2) * (1 + (-x^2/2! - x*y ) + ( x^4/4! + x*y^3/3! ) + (-x^6/6! + 8*x^4*y^2/(4!*2!) + 8*x^3*y^3/(3!*3!) - x*y^5/5! ) + ( x^8/8! - 24*x^6*y^2/(6!*2!) - 24*x^3*y^5/(3!*5!) + x*y^7/7! ) + (-x^10/10! + 48*x^8*y^2/(8!*2!) - 576*x^6*y^4/(6!*4!) - 576*x^5*y^5/(5!*5!) + 48*x^3*y^7/(3!*7!) - x*y^9/9! ) + ( x^12/12! - 80*x^10*y^2/(10!*2!) + 3200*x^8*y^4/(8!*4!) + 3200*x^5*y^7/(5!*7!) - 80*x^3*y^9/(3!*9!) + x*y^11/11! ) + (-x^14/14! + 120*x^12*y^2/(12!*2!) - 10240*x^10*y^4/(10!*4!) + 160768*x^8*y^6/(8!*6!) + 160768*x^7*y^7/(7!*7!) - 10240*x^5*y^9/(5!*9!) + 120*x^3*y^11/(3!*11!) - x*y^13/13! ) + ...).
The e.g.f. of A326802 begins
D(x,y) = sqrt(1/2) * (1 + (-x^2/2! + x*y ) + ( x^4/4! - x*y^3/3! ) + (-x^6/6! + 8*x^4*y^2/(4!*2!) - 8*x^3*y^3/(3!*3!) + x*y^5/5! ) + ( x^8/8! - 24*x^6*y^2/(6!*2!) + 24*x^3*y^5/(3!*5!) - x*y^7/7! ) + (-x^10/10! + 48*x^8*y^2/(8!*2!) - 576*x^6*y^4/(6!*4!) + 576*x^5*y^5/(5!*5!) - 48*x^3*y^7/(3!*7!) + x*y^9/9! ) + ( x^12/12! - 80*x^10*y^2/(10!*2!) + 3200*x^8*y^4/(8!*4!) - 3200*x^5*y^7/(5!*7!) + 80*x^3*y^9/(3!*9!) - x*y^11/11! ) + (-x^14/14! + 120*x^12*y^2/(12!*2!) - 10240*x^10*y^4/(10!*4!) + 160768*x^8*y^6/(8!*6!) - 160768*x^7*y^7/(7!*7!) + 10240*x^5*y^9/(5!*9!) - 120*x^3*y^11/(3!*11!) + x*y^13/13! ) + ...).
The e.g.f. of A326552 begins
SS(x*y) = (x*y) - 8*(x*y)^3/3!^2 + 576*(x*y)^5/5!^2 - 160768*(x*y)^7/7!^2 + 123535360*(x*y)^9/9!^2 - 212713734144*(x*y)^11/11!^2 + 716196297048064*(x*y)^13/13!^2 - 4280584942657732608*(x*y)^15/15!^2 + 42250703121584165486592*(x*y)^17/17!^2 - 651154631135458759089848320*(x*y)^19/19!^2 + 14983590319172065236171175755776*(x*y)^21/21!^2 + ... + A326552(n)*(x*y)^(2*n-1)/(2*n-1)! + ...
such that
SS(x*y) = Dx*Dy - Cx*Cy - Sx*Sy.
The e.g.f. of A326551 begins
CC(x*y) = 1 - 2*(x*y)^2/2!^2 + 56*(x*y)^4/4!^2 - 8336*(x*y)^6/6!^2 + 3985792*(x*y)^8/8!^2 - 4679517952*(x*y)^10/10!^2 + 11427218287616*(x*y)^12/12!^2 - 51793067942397952*(x*y)^14/14!^2 + 400951893341645930496*(x*y)^16/16!^2 - 4975999084909976839454720*(x*y)^18/18!^2 + 94178912073481319162642169856*(x*y)^20/20!^2 -+ ... + A326551(n)*(x*y)^(2*n)/(2*n)! + ...
such that
CC(x*y)^2 = (Cx*Dy + Cy*Dx)^2 + (Sx*Dy + Sy*Dx)^2 + (Sx*Cy - Sy*Cx)^2,
and CC(x*y)^2 + SS(x*y)^2 = 1.
PROG
(PARI)
{TSx(n, k) = my(Cx=1, Sx=x, Dx=1, Cy=1, Sy=y, Dy=1);
for(i=0, 2*n+1,
Sx = intformal( Cx*Dy + Cy*Dx, x) + O(x^(2*n+2));
Cx = sqrt(1/2) - intformal( Sx*Dy + Sy*Dx, x);
Dx = sqrt(1/2) - intformal( Sx*Cy - Sy*Cx, x);
Sy = intformal( Cy*Dx + Cx*Dy, y) + O(y^(2*n+2));
Cy = sqrt(1/2) - intformal( Sy*Dx + Sx*Dy, y);
Dy = sqrt(1/2) - intformal( Sy*Cx - Sx*Cy, y);
);
round( (2*n-2*k+1)!*(2*k)! * polcoeff( polcoeff(Sx, 2*n-2*k+1, x), 2*k, y) )}
for(n=0, 10, for(k=0, n, print1( TSx(n, k), ", ")); print(""))
CROSSREFS
Cf. A326801 (Cx), A326802 (Dx), A326803 (central terms).
Cf. A326551 (CC), A326552 (SS), A326797.
KEYWORD
sign,tabl
AUTHOR
Paul D. Hanna, Jul 27 2019
STATUS
approved
A326797
Consider the e.g.f. A(x,y) = Sum_{n>=0} Sum_{k=0..n} T(n,k) * x^(2*n-2*k+1) * y^(2*k) / (2*n+1)! and related functions B(x,y) and C(x,y), as defined in the Formula section. Sequence gives the triangular array of coefficients T(n,k) (n>=0, 0<=k<=n) of A(x,y).
+10
7
1, -1, -3, 1, -30, 5, -1, 315, 525, -7, 1, -1260, 18270, -2940, 9, -1, 3465, -496650, -695310, 10395, -11, 1, -7722, 4279275, -52144092, 7702695, -28314, 13, -1, 15015, -22387365, 2701093395, 3472834365, -49252203, 65065, -15, 1, -26520, 86786700, -40541436936, 454101489270, -63707972328, 225645420, -132600, 17
(list; table; graph; refs; listen; history; text; internal format)
OFFSET
0,3
COMMENTS
The e.g.f. of this triangle is equivalent to the e.g.f. of triangle A326800, where T(n,k) = A326800(n,k) * binomial(2*n+1, 2*k).
The e.g.f. A(x,y) at y = x is described by A326794.
LINKS
Paul D. Hanna, Table of n, a(n) for n = 0..1890
FORMULA
The e.g.f. Ax = A(x,y) = Sum_{n>=0} Sum_{k=0..n} T(n,k)*x^(2*n-2*k+1)*y^(2*k)/(2*n+1)! and related functions Bx = B(x,y), Cx = C(x,y), Ay = A(y,x), By = B(y,x), and Cy = C(y,x) satisfy the following relations.
DEFINITION.
(1a) Ax = 0 + Integral Bx*Cy - Cx*By dx,
(1b) Bx = 1 + Integral Cx*Ay - Ax*Cy dx,
(1c) Cx = 0 + Integral Ax*By - Bx*Ay dx.
(2a) Ay = 0 + Integral By*Cx - Cy*Bx dy,
(2b) By = 0 + Integral Cy*Ax - Ay*Cx dy,
(2c) Cy = 1 + Integral Ay*Bx - By*Ax dy.
IDENTITIES.
(3a) Ax^2 + Bx^2 + Cx^2 = 1.
(3b) Ay^2 + By^2 + Cy^2 = 1.
(4a) (Ax*Ay + Bx*By + Cx*Cy)^2 + (d/dx Ax)^2 + (d/dx Bx)^2 + (d/dx Cx)^2 = 1.
(4b) (Ax*Ay + Bx*By + Cx*Cy)^2 + (d/dy Ay)^2 + (d/dy By)^2 + (d/dy Cy)^2 = 1.
(5a) Ax*(d/dx Ax) + Bx*(d/dx Bx) + Cx*(d/dx Cx) = 0.
(5b) Ay*(d/dy Ay) + By*(d/dy By) + Cy*(d/dy Cy) = 0.
(5c) Ax*(d/dy Ay) + Bx*(d/dy By) + Cx*(d/dy Cy) = 0.
(5d) Ay*(d/dx Ax) + By*(d/dx Bx) + Cy*(d/dx Cx) = 0.
(5e) Ax*(d/dy Ax) + Bx*(d/dy Bx) + Cx*(d/dy Cx) = 0.
(5f) Ay*(d/dx Ay) + By*(d/dx By) + Cy*(d/dx Cy) = 0.
RELATED FUNCTIONS.
(6a) SS(x*y) = Ax*Ay + Bx*By + Cx*Cy.
(6b) d/dx SS(x*y) = Ax*(d/dx Ay) + Bx*(d/dx By) + Cx*(d/dx Cy).
(6c) d/dy SS(x*y) = Ay*(d/dy Ax) + By*(d/dy Bx) + Cy*(d/dy Cx).
(7a) CC(x*y)^2 = (Bx*Cy - Cx*By)^2 + (Cx*Ay - Ax*Cy)^2 + (Ax*By - Bx*Ay)^2.
(7b) CC(x*y)^2 = (d/dx Ax)^2 + (d/dx Bx)^2 + (d/dx Cx)^2.
(7c) CC(x*y)^2 = (d/dy Ay)^2 + (d/dy By)^2 + (d/dy Cy)^2.
In the above, CC(x) and SS(x) are the e.g.f.s of A326551 and A326552 defined by
(8a) CC(x*y)^2 + SS(x*y)^2 = 1,
(8b) SS(x*y) = Integral CC(x*y) * (Integral CC(x*y) dy) dx,
(8c) CC(x*y) = 1 - Integral SS(x*y) * (Integral CC(x*y) dy) dx,
(8d) SS(x*y) = sin( Integral Integral CC(x*y) dx dy ),
(8e) CC(x*y) = cos( Integral Integral CC(x*y) dx dy ).
OTHER RELATIONS.
(9a) Ay = Ax*SS(x*y) - Bx*(d/dx Cx) + Cx*(d/dx Bx).
(9b) By = Bx*SS(x*y) - Cx*(d/dx Ax) + Ax*(d/dx Cx).
(9c) Cy = Cx*SS(x*y) - Ax*(d/dx Bx) + Bx*(d/dx Ax).
(9d) Ax = Ay*SS(x*y) - By*(d/dy Cy) + Cy*(d/dy By).
(9e) Bx = By*SS(x*y) - Cy*(d/dy Ay) + Ay*(d/dy Cy).
(9f) Cx = Cy*SS(x*y) - Ay*(d/dy By) + By*(d/dy Ay).
DERIVATIVES.
(10a) d/dx Ax = Bx*Cy - Cx*By.
(10b) d/dx Bx = Cx*Ay - Ax*Cy.
(10c) d/dx Cx = Ax*By - Bx*Ay.
(10d) d/dy Ay = By*Cx - Cy*Bx.
(10e) d/dy By = Cy*Ax - Ay*Cx.
(10f) d/dy Cy = Ay*Bx - By*Ax.
VECTOR FORM.
Set radial vectors Vx = [Ax,Bx,Cx] and Vy = [Ay,By,Cy], then we can write the above relations in compact form using cross (X) products and dot (*) products.
(1) Vx = [0,1,0] + Integral Vx X Vy dx.
(2) Vy = [0,0,1] + Integral Vy X Vx dy.
(3a) Vx * Vx = 1.
(3b) Vy * Vy = 1.
(4a) (Vx * Vy)^2 + (d/dx Vx) * (d/dx Vx) = 1.
(4b) (Vx * Vy)^2 + (d/dy Vy) * (d/dy Vy) = 1.
(5a) Vx * (d/dx Vx) = 0.
(5b) Vy * (d/dy Vy) = 0.
(5c) Vx * (d/dy Vy) = 0.
(5d) Vy * (d/dx Vx) = 0.
(5e) Vx * (d/dy Vx) = 0.
(5f) Vy * (d/dx Vy) = 0.
(6a) SS(x*y) = Vx * Vy.
(6b) d/dx SS(x*y) = Vx * (d/dx Vy).
(6c) d/dy SS(x*y) = Vy * (d/dy Vx).
(7) CC(x*y)^2 = (Vx X Vy) * (Vx X Vy) = 1 - (Vx * Vy)^2.
(9a-c) Vy = Vx*SS(x*y) - Vx X (d/dx Vx) because Vx X (Vx X Vy) = Vx*(Vx * Vy) - Vy.
(9d-f) Vx = Vy*SS(x*y) - Vy X (d/dy Vy) because Vy X (Vy X Vx) = Vy*(Vx * Vy) - Vx.
(10a-c) d/dx Vx = Vx X Vy.
(10d-f) d/dy Vy = Vy X Vx.
EXAMPLE
E.g.f.: A(x,y) = x + (-1*x^3 - 3*x*y^2)/3! + (1*x^5 - 30*x^3*y^2 + 5*x*y^4)/5! + (-1*x^7 + 315*x^5*y^2 + 525*x^3*y^4 - 7*x*y^6)/7! + (1*x^9 - 1260*x^7*y^2 + 18270*x^5*y^4 - 2940*x^3*y^6 + 9*x*y^8)/9! + (-1*x^11 + 3465*x^9*y^2 - 496650*x^7*y^4 - 695310*x^5*y^6 + 10395*x^3*y^8 - 11*x*y^10)/11! + (1*x^13 - 7722*x^11*y^2 + 4279275*x^9*y^4 - 52144092*x^7*y^6 + 7702695*x^5*y^8 - 28314*x^3*y^10 + 13*x*y^12)/13! + (-1*x^15 + 15015*x^13*y^2 - 22387365*x^11*y^4 + 2701093395*x^9*y^6 + 3472834365*x^7*y^8 - 49252203*x^5*y^10 + 65065*x^3*y^12 - 15*x*y^14)/15! + (1*x^17 - 26520*x^15*y^2 + 86786700*x^13*y^4 - 40541436936*x^11*y^6 + 454101489270*x^9*y^8 - 63707972328*x^7*y^10 + 225645420*x^5*y^12 - 132600*x^3*y^14 + 17*x*y^16)/17! +(-1*x^19 + 43605*x^17*y^2 - 274362660*x^15*y^4 + 345219726852*x^13*y^6 - 38308692031038*x^11*y^8 - 46821734704602*x^9*y^10 + 641122349868*x^7*y^12 - 823087980*x^5*y^14 + 247095*x^3*y^16 - 19*x*y^18)/19! + ...
such that
. A(x,y) = 0 + Integral B(x,y)*C(y,x) - C(x,y)*B(y,x) dx,
. A(y,x) = 0 + Integral B(y,x)*C(x,y) - C(y,x)*B(x,y) dy,
where B(x,y) and C(x,y) satisfy
. A(x,y)^2 + B(x,y)^2 + C(x,y)^2 = 1.
TRIANGLE.
This triangle of coefficients T(n,k) of x^(2*n-2*k+1) * y^(2*k) / (2*n+1)! in A(x,y) begins
1;
-1, -3;
1, -30, 5;
-1, 315, 525, -7;
1, -1260, 18270, -2940, 9;
-1, 3465, -496650, -695310, 10395, -11;
1, -7722, 4279275, -52144092, 7702695, -28314, 13;
-1, 15015, -22387365, 2701093395, 3472834365, -49252203, 65065, -15;
1, -26520, 86786700, -40541436936, 454101489270, -63707972328, 225645420, -132600, 17;
-1, 43605, -274362660, 345219726852, -38308692031038, -46821734704602, 641122349868, -823087980, 247095, -19;
1, -67830, 747317025, -2065684781160, 887850774580770, -9453938937390948, 1282451118838890, -4426467388200, 2540877885, -429590, 21; ...
RELATED TRIANGLE.
A related triangle (A326800), formed from coefficients of x^(2*n-2*k+1) * y^(2*k) / ((2*n-2*k+1)!*(2*k)!) in e.g.f. A(x,y), begins
1;
-1, -1;
1, -3, 1;
-1, 15, 15, -1;
1, -35, 145, -35, 1;
-1, 63, -1505, -1505, 63, -1;
1, -99, 5985, -30387, 5985, -99, 1;
-1, 143, -16401, 539679, 539679, -16401, 143, -1;
1, -195, 36465, -3275811, 18679617, -3275811, 36465, -195, 1; ...
RELATED FUNCTIONS.
B(x,y) = 1 + (-1*x^2)/2! + (1*x^4)/4! + (-1*x^6 + 120*x^4*y^2)/6! + (1*x^8 - 672*x^6*y^2)/8! + (-1*x^10 + 2160*x^8*y^2 - 120960*x^6*y^4)/10! + (1*x^12 - 5280*x^10*y^2 + 1584000*x^8*y^4)/12! + (-1*x^14 + 10920*x^12*y^2 - 10250240*x^10*y^4 + 482786304*x^8*y^6)/14! + (1*x^16 - 20160*x^14*y^2 + 45427200*x^12*y^4 - 11480268800*x^10*y^6)/16! + ...
such that
. B(x,y) = 1 + Integral C(x,y)*A(y,x) - A(x,y)*C(y,x) dx,
. B(y,x) = 0 + Integral C(y,x)*A(x,y) - A(y,x)*C(x,y) dy.
C(x,y) = (2*x*y)/2! + (-4*x*y^3)/4! + (-160*x^3*y^3 + 6*x*y^5)/6! + (1344*x^3*y^5 - 8*x*y^7)/8! + (145152*x^5*y^5 - 5760*x^3*y^7 + 10*x*y^9)/10! + (-2534400*x^5*y^7 + 17600*x^3*y^9 - 12*x*y^11)/12! + (-551755776*x^7*y^7 + 20500480*x^5*y^9 - 43680*x^3*y^11 + 14*x*y^13)/14! + (16400384000*x^7*y^9 - 109025280*x^5*y^11 + 94080*x^3*y^13 - 16*x*y^15)/16! + ...
such that
. C(x,y) = 0 + Integral A(x,y)*B(y,x) - B(x,y)*A(y,x) dx,
. C(y,x) = 1 + Integral A(y,x)*B(x,y) - B(y,x)*A(x,y) dy.
CC(x) = 1 - 2*x^2/2!^2 + 56*x^4/4!^2 - 8336*x^6/6!^2 + 3985792*x^8/8!^2 - 4679517952*x^10/10!^2 + 11427218287616*x^12/12!^2 - 51793067942397952*x^14/14!^2 + 400951893341645930496*x^16/16!^2 + ... + A326551(n)*x^(2*n)/(2*n)!^2 + ...
such that
. CC(x*y) = sqrt( (Bx*Cy - Cx*By)^2 + (Cx*Ay - Ax*Cy)^2 + (Ax*By - Bx*Ay)^2 ).
SS(x) = x - 8*x^3/3!^2 + 576*x^5/5!^2 - 160768*x^7/7!^2 + 123535360*x^9/9!^2 - 212713734144*x^11/11!^2 + 716196297048064*x^13/13!^2 - 4280584942657732608*x^15/15!^2 + 42250703121584165486592*x^17/17!^2 + ... + A326552(n)*x^(2*n+1)/(2*n+1)!^2 + ...
such that SS(x*y) = Ax*Ay + Bx*By + Cx*Cy.
PROG
(PARI) {TAx(n, k) = my(Ax=1, Bx=x, Cx=1, Ay=1, By=y, Cy=1);
for(i=0, 2*n+1,
Ax = 0 + intformal( Bx*Cy - Cx*By, x) + O(x^(2*n+2));
Bx = 1 + intformal( Cx*Ay - Ax*Cy, x) + O(x^(2*n+2));
Cx = 0 + intformal( Ax*By - Bx*Ay, x) + O(x^(2*n+2));
Ay = 0 + intformal( By*Cx - Cy*Bx, y) + O(y^(2*n+2));
By = 0 + intformal( Cy*Ax - Ay*Cx, y) + O(y^(2*n+2));
Cy = 1 + intformal( Ay*Bx - By*Ax, y) + O(y^(2*n+2));
);
(2*n+1)! * polcoeff( polcoeff(Ax, 2*n-2*k+1, x), 2*k, y)}
for(n=0, 10, for(k=0, n, print1( TAx(n, k), ", ")); print(""))
CROSSREFS
Cf. A326798 (B), A326799 (C), A326800.
Cf. A326794 (row sums), A326551 (CC), A326552 (SS).
KEYWORD
sign,tabl,look
AUTHOR
Paul D. Hanna, Aug 03 2019
STATUS
approved
A326798
Consider the e.g.f. B(x,y) = Sum_{n>=0} Sum_{k=0..floor(n/2)} T(n,k) * x^(2*n-2*k) * y^(2*k) / (2*n)! and related functions A(x,y) and C(x,y), as defined in the Formula section. Sequence gives the triangular array of coefficients T(n,k) (n>=0, 0<=k<=floor(n/2)) of B(x,y).
+10
6
1, -1, 1, 0, -1, 120, 1, -672, 0, -1, 2160, -120960, 1, -5280, 1584000, 0, -1, 10920, -10250240, 482786304, 1, -20160, 45427200, -11480268800, 0, -1, 34272, -157651200, 124816613376, -5405660282880, 1, -54720, 460158720, -875447623680, 203526629130240, 0
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,6
COMMENTS
The e.g.f. B(x,y) at y = x is described by A326795.
LINKS
Paul D. Hanna, Table of n, a(n) for n = 0..960
FORMULA
The e.g.f. Bx = B(x,y) = Sum_{n>=0} Sum_{k=0..floor(n/2)} T(n,k)*x^(2*n-2*k)*y^(2*k)/(2*n)! and related functions Ax = A(x,y), Cx = C(x,y), Ay = A(y,x), By = B(y,x), and Cy = C(y,x) satisfy the following relations.
DEFINITION.
(1a) Ax = 0 + Integral Bx*Cy - Cx*By dx,
(1b) Bx = 1 + Integral Cx*Ay - Ax*Cy dx,
(1c) Cx = 0 + Integral Ax*By - Bx*Ay dx.
(2a) Ay = 0 + Integral By*Cx - Cy*Bx dy,
(2b) By = 0 + Integral Cy*Ax - Ay*Cx dy,
(2c) Cy = 1 + Integral Ay*Bx - By*Ax dy.
IDENTITIES.
(3a) Ax^2 + Bx^2 + Cx^2 = 1.
(3b) Ay^2 + By^2 + Cy^2 = 1.
(4a) (Ax*Ay + Bx*By + Cx*Cy)^2 + (d/dx Ax)^2 + (d/dx Bx)^2 + (d/dx Cx)^2 = 1.
(4b) (Ax*Ay + Bx*By + Cx*Cy)^2 + (d/dy Ay)^2 + (d/dy By)^2 + (d/dy Cy)^2 = 1.
(5a) Ax*(d/dx Ax) + Bx*(d/dx Bx) + Cx*(d/dx Cx) = 0.
(5b) Ay*(d/dy Ay) + By*(d/dy By) + Cy*(d/dy Cy) = 0.
(5c) Ax*(d/dy Ay) + Bx*(d/dy By) + Cx*(d/dy Cy) = 0.
(5d) Ay*(d/dx Ax) + By*(d/dx Bx) + Cy*(d/dx Cx) = 0.
(5e) Ax*(d/dy Ax) + Bx*(d/dy Bx) + Cx*(d/dy Cx) = 0.
(5f) Ay*(d/dx Ay) + By*(d/dx By) + Cy*(d/dx Cy) = 0.
RELATED FUNCTIONS.
(6a) SS(x*y) = Ax*Ay + Bx*By + Cx*Cy.
(6b) d/dx SS(x*y) = Ax*(d/dx Ay) + Bx*(d/dx By) + Cx*(d/dx Cy).
(6c) d/dy SS(x*y) = Ay*(d/dy Ax) + By*(d/dy Bx) + Cy*(d/dy Cx).
(7a) CC(x*y)^2 = (Bx*Cy - Cx*By)^2 + (Cx*Ay - Ax*Cy)^2 + (Ax*By - Bx*Ay)^2.
(7b) CC(x*y)^2 = (d/dx Ax)^2 + (d/dx Bx)^2 + (d/dx Cx)^2.
(7c) CC(x*y)^2 = (d/dy Ay)^2 + (d/dy By)^2 + (d/dy Cy)^2.
In the above, CC(x) and SS(x) are the e.g.f.s of A326551 and A326552 defined by
(8a) CC(x*y)^2 + SS(x*y)^2 = 1,
(8b) SS(x*y) = Integral CC(x*y) * (Integral CC(x*y) dy) dx,
(8c) CC(x*y) = 1 - Integral SS(x*y) * (Integral CC(x*y) dy) dx,
(8d) SS(x*y) = sin( Integral Integral CC(x*y) dx dy ),
(8e) CC(x*y) = cos( Integral Integral CC(x*y) dx dy ).
OTHER RELATIONS..
(9a) Ay = Ax*SS(x*y) - Bx*(d/dx Cx) + Cx*(d/dx Bx).
(9b) By = Bx*SS(x*y) - Cx*(d/dx Ax) + Ax*(d/dx Cx).
(9c) Cy = Cx*SS(x*y) - Ax*(d/dx Bx) + Bx*(d/dx Ax).
(9d) Ax = Ay*SS(x*y) - By*(d/dy Cy) + Cy*(d/dy By).
(9e) Bx = By*SS(x*y) - Cy*(d/dy Ay) + Ay*(d/dy Cy).
(9f) Cx = Cy*SS(x*y) - Ay*(d/dy By) + By*(d/dy Ay).
DERIVATIVES.
(10a) d/dx Ax = Bx*Cy - Cx*By.
(10b) d/dx Bx = Cx*Ay - Ax*Cy.
(10c) d/dx Cx = Ax*By - Bx*Ay.
(10d) d/dy Ay = By*Cx - Cy*Bx.
(10e) d/dy By = Cy*Ax - Ay*Cx.
(10f) d/dy Cy = Ay*Bx - By*Ax.
EXAMPLE
E.g.f.: B(x,y) = 1 + (-1*x^2)/2! + (1*x^4)/4! + (-1*x^6 + 120*x^4*y^2)/6! + (1*x^8 - 672*x^6*y^2)/8! + (-1*x^10 + 2160*x^8*y^2 - 120960*x^6*y^4)/10! + (1*x^12 - 5280*x^10*y^2 + 1584000*x^8*y^4)/12! + (-1*x^14 + 10920*x^12*y^2 - 10250240*x^10*y^4 + 482786304*x^8*y^6)/14! + (1*x^16 - 20160*x^14*y^2 + 45427200*x^12*y^4 - 11480268800*x^10*y^6)/16! + (-1*x^18 + 34272*x^16*y^2 - 157651200*x^14*y^4 + 124816613376*x^12*y^6 - 5405660282880*x^10*y^8)/18! + (1*x^20 - 54720*x^18*y^2 + 460158720*x^16*y^4 - 875447623680*x^14*y^6 + 203526629130240*x^12*y^8)/20! + ...
such that
. B(x,y) = 1 + Integral C(x,y)*A(y,x) - A(x,y)*C(y,x) dx,
. B(y,x) = 0 + Integral C(y,x)*A(x,y) - A(y,x)*C(x,y) dy,
where A(x,y) and C(x,y) satisfy
. A(x,y)^2 + B(x,y)^2 + C(x,y)^2 = 1.
TRIANGLE.
This triangle of coefficients T(n,k) of x^(2*n-2*k)*y^(2*k)/(2*n)! in B(x,y) begins
1;
-1;
1, 0;
-1, 120;
1, -672, 0;
-1, 2160, -120960;
1, -5280, 1584000, 0;
-1, 10920, -10250240, 482786304;
1, -20160, 45427200, -11480268800, 0;
-1, 34272, -157651200, 124816613376, -5405660282880;
1, -54720, 460158720, -875447623680, 203526629130240, 0;
-1, 83160, -1179763200, 4585597986816, -3340908170772480, 137550485329281024;
1, -121440, 2733857280, -19438470610944, 34039224247615488, -7523050148723687424, 0; ...
RELATED FUNCTIONS.
A(x,y) = x + (-1*x^3 - 3*x*y^2)/3! + (1*x^5 - 30*x^3*y^2 + 5*x*y^4)/5! + (-1*x^7 + 315*x^5*y^2 + 525*x^3*y^4 - 7*x*y^6)/7! + (1*x^9 - 1260*x^7*y^2 + 18270*x^5*y^4 - 2940*x^3*y^6 + 9*x*y^8)/9! + (-1*x^11 + 3465*x^9*y^2 - 496650*x^7*y^4 - 695310*x^5*y^6 + 10395*x^3*y^8 - 11*x*y^10)/11! + (1*x^13 - 7722*x^11*y^2 + 4279275*x^9*y^4 - 52144092*x^7*y^6 + 7702695*x^5*y^8 - 28314*x^3*y^10 + 13*x*y^12)/13! + (-1*x^15 + 15015*x^13*y^2 - 22387365*x^11*y^4 + 2701093395*x^9*y^6 + 3472834365*x^7*y^8 - 49252203*x^5*y^10 + 65065*x^3*y^12 - 15*x*y^14)/15! + ...
such that
. A(x,y) = 0 + Integral B(x,y)*C(y,x) - C(x,y)*B(y,x) dx,
. A(y,x) = 0 + Integral B(y,x)*C(x,y) - C(y,x)*B(x,y) dy.
C(x,y) = (2*x*y)/2! + (-4*x*y^3)/4! + (-160*x^3*y^3 + 6*x*y^5)/6! + (1344*x^3*y^5 - 8*x*y^7)/8! + (145152*x^5*y^5 - 5760*x^3*y^7 + 10*x*y^9)/10! + (-2534400*x^5*y^7 + 17600*x^3*y^9 - 12*x*y^11)/12! + (-551755776*x^7*y^7 + 20500480*x^5*y^9 - 43680*x^3*y^11 + 14*x*y^13)/14! + (16400384000*x^7*y^9 - 109025280*x^5*y^11 + 94080*x^3*y^13 - 16*x*y^15)/16! + ...
such that
. C(x,y) = 0 + Integral A(x,y)*B(y,x) - B(x,y)*A(y,x) dx,
. C(y,x) = 1 + Integral A(y,x)*B(x,y) - B(y,x)*A(x,y) dy.
CC(x) = 1 - 2*x^2/2!^2 + 56*x^4/4!^2 - 8336*x^6/6!^2 + 3985792*x^8/8!^2 - 4679517952*x^10/10!^2 + 11427218287616*x^12/12!^2 - 51793067942397952*x^14/14!^2 + 400951893341645930496*x^16/16!^2 + ... + A326551(n)*x^(2*n)/(2*n)!^2 + ...
such that
. CC(x*y) = sqrt( (Bx*Cy - Cx*By)^2 + (Cx*Ay - Ax*Cy)^2 + (Ax*By - Bx*Ay)^2 ).
SS(x) = x - 8*x^3/3!^2 + 576*x^5/5!^2 - 160768*x^7/7!^2 + 123535360*x^9/9!^2 - 212713734144*x^11/11!^2 + 716196297048064*x^13/13!^2 - 4280584942657732608*x^15/15!^2 + 42250703121584165486592*x^17/17!^2 + ... + A326552(n)*x^(2*n+1)/(2*n+1)!^2 + ...
such that SS(x*y) = Ax*Ay + Bx*By + Cx*Cy.
PROG
(PARI) {TBx(n, k) = my(Ax=x, Bx=1, Cx=x, Ay=y, By=y, Cy=1);
for(i=0, 2*n+1,
Ax = 0 + intformal( Bx*Cy - Cx*By, x) + O(x^(2*n+2));
Bx = 1 + intformal( Cx*Ay - Ax*Cy, x) + O(x^(2*n+2));
Cx = 0 + intformal( Ax*By - Bx*Ay, x) + O(x^(2*n+2));
Ay = 0 + intformal( By*Cx - Cy*Bx, y) + O(y^(2*n+2));
By = 0 + intformal( Cy*Ax - Ay*Cx, y) + O(y^(2*n+2));
Cy = 1 + intformal( Ay*Bx - By*Ax, y) + O(y^(2*n+2));
);
(2*n)! * polcoeff( polcoeff(Bx, 2*n-2*k, x), 2*k, y)}
for(n=0, 10, for(k=0, n\2, print1( TBx(n, k), ", ")); print(""))
CROSSREFS
Cf. A326797 (A), A326799 (C).
Cf. A326795 (row sums), A326551 (CC), A326552 (SS).
KEYWORD
sign,tabf
AUTHOR
Paul D. Hanna, Aug 03 2019
STATUS
approved
A326799
Consider the e.g.f. C(x,y) = Sum_{n>=0} Sum_{k=0..n} T(n,k) * x^(2*n-2*k+1) * y^(2*k+1) / (2*n+2)! and related functions A(x,y) and B(x,y), as defined in the Formula section. Sequence gives the triangular array of coefficients T(n,k) (n>=0, 0<=k<=n) of C(x,y).
+10
6
2, 0, -4, 0, -160, 6, 0, 0, 1344, -8, 0, 0, 145152, -5760, 10, 0, 0, 0, -2534400, 17600, -12, 0, 0, 0, -551755776, 20500480, -43680, 14, 0, 0, 0, 0, 16400384000, -109025280, 94080, -16, 0, 0, 0, 0, 6006289203200, -213971337216, 441423360, -182784, 18
(list; table; graph; refs; listen; history; text; internal format)
OFFSET
0,1
COMMENTS
The e.g.f. C(x,y) at y = x is described by A326796.
LINKS
Paul D. Hanna, Table of n, a(n) for n = 0..1890
FORMULA
The e.g.f. Cx = C(x,y) = Sum_{n>=0} Sum_{k=0..n} T(n,k)*x^(2*n-2*k+1)*y^(2*k+1)/(2*n+2)! and related functions Ax = A(x,y), Bx = B(x,y), Ay = A(y,x), By = B(y,x), and Cy = C(y,x) satisfy the following relations.
DEFINITION.
(1a) Ax = 0 + Integral Bx*Cy - Cx*By dx,
(1b) Bx = 1 + Integral Cx*Ay - Ax*Cy dx,
(1c) Cx = 0 + Integral Ax*By - Bx*Ay dx.
(2a) Ay = 0 + Integral By*Cx - Cy*Bx dy,
(2b) By = 0 + Integral Cy*Ax - Ay*Cx dy,
(2c) Cy = 1 + Integral Ay*Bx - By*Ax dy.
IDENTITIES.
(3a) Ax^2 + Bx^2 + Cx^2 = 1.
(3b) Ay^2 + By^2 + Cy^2 = 1.
(4a) (Ax*Ay + Bx*By + Cx*Cy)^2 + (d/dx Ax)^2 + (d/dx Bx)^2 + (d/dx Cx)^2 = 1.
(4b) (Ax*Ay + Bx*By + Cx*Cy)^2 + (d/dy Ay)^2 + (d/dy By)^2 + (d/dy Cy)^2 = 1.
(5a) Ax*(d/dx Ax) + Bx*(d/dx Bx) + Cx*(d/dx Cx) = 0.
(5b) Ay*(d/dy Ay) + By*(d/dy By) + Cy*(d/dy Cy) = 0.
(5c) Ax*(d/dy Ay) + Bx*(d/dy By) + Cx*(d/dy Cy) = 0.
(5d) Ay*(d/dx Ax) + By*(d/dx Bx) + Cy*(d/dx Cx) = 0.
(5e) Ax*(d/dy Ax) + Bx*(d/dy Bx) + Cx*(d/dy Cx) = 0.
(5f) Ay*(d/dx Ay) + By*(d/dx By) + Cy*(d/dx Cy) = 0.
RELATED FUNCTIONS.
(6a) SS(x*y) = Ax*Ay + Bx*By + Cx*Cy.
(6b) d/dx SS(x*y) = Ax*(d/dx Ay) + Bx*(d/dx By) + Cx*(d/dx Cy).
(6c) d/dy SS(x*y) = Ay*(d/dy Ax) + By*(d/dy Bx) + Cy*(d/dy Cx).
(7a) CC(x*y)^2 = (Bx*Cy - Cx*By)^2 + (Cx*Ay - Ax*Cy)^2 + (Ax*By - Bx*Ay)^2.
(7b) CC(x*y)^2 = (d/dx Ax)^2 + (d/dx Bx)^2 + (d/dx Cx)^2.
(7c) CC(x*y)^2 = (d/dy Ay)^2 + (d/dy By)^2 + (d/dy Cy)^2.
In the above, CC(x) and SS(x) are the e.g.f.s of A326551 and A326552 defined by
(8a) CC(x*y)^2 + SS(x*y)^2 = 1,
(8b) SS(x*y) = Integral CC(x*y) * (Integral CC(x*y) dy) dx,
(8c) CC(x*y) = 1 - Integral SS(x*y) * (Integral CC(x*y) dy) dx,
(8d) SS(x*y) = sin( Integral Integral CC(x*y) dx dy ),
(8e) CC(x*y) = cos( Integral Integral CC(x*y) dx dy ).
OTHER RELATIONS.
(9a) Ay = Ax*SS(x*y) - Bx*(d/dx Cx) + Cx*(d/dx Bx).
(9b) By = Bx*SS(x*y) - Cx*(d/dx Ax) + Ax*(d/dx Cx).
(9c) Cy = Cx*SS(x*y) - Ax*(d/dx Bx) + Bx*(d/dx Ax).
(9d) Ax = Ay*SS(x*y) - By*(d/dy Cy) + Cy*(d/dy By).
(9e) Bx = By*SS(x*y) - Cy*(d/dy Ay) + Ay*(d/dy Cy).
(9f) Cx = Cy*SS(x*y) - Ay*(d/dy By) + By*(d/dy Ay).
DERIVATIVES.
(10a) d/dx Ax = Bx*Cy - Cx*By.
(10b) d/dx Bx = Cx*Ay - Ax*Cy.
(10c) d/dx Cx = Ax*By - Bx*Ay.
(10d) d/dy Ay = By*Cx - Cy*Bx.
(10e) d/dy By = Cy*Ax - Ay*Cx.
(10f) d/dy Cy = Ay*Bx - By*Ax.
EXAMPLE
E.g.f.: C(x,y) = (2*x*y)/2! + (-4*x*y^3)/4! + (-160*x^3*y^3 + 6*x*y^5)/6! + (1344*x^3*y^5 - 8*x*y^7)/8! + (145152*x^5*y^5 - 5760*x^3*y^7 + 10*x*y^9)/10! + (-2534400*x^5*y^7 + 17600*x^3*y^9 - 12*x*y^11)/12! + (-551755776*x^7*y^7 + 20500480*x^5*y^9 - 43680*x^3*y^11 + 14*x*y^13)/14! + (16400384000*x^7*y^9 - 109025280*x^5*y^11 + 94080*x^3*y^13 - 16*x*y^15)/16! + (6006289203200*x^9*y^9 - 213971337216*x^7*y^11 + 441423360*x^5*y^13 - 182784*x^3*y^15 + 18*x*y^17)/18! + (-271368838840320*x^9*y^11 + 1750895247360*x^7*y^13 - 1472507904*x^5*y^15 + 328320*x^3*y^17 - 20*x*y^19)/20! + (-150055074904670208*x^11*y^11 + 5196968265646080*x^9*y^13 - 10481366827008*x^7*y^15 + 4247147520*x^5*y^17 - 554400*x^3*y^19 + 22*x*y^21)/22! + ...
such that
. C(x,y) = 0 + Integral A(x,y)*B(y,x) - B(x,y)*A(y,x) dx,
. C(y,x) = 1 + Integral A(y,x)*B(x,y) - B(y,x)*A(x,y) dy,
where A(x,y) and B(x,y) satisfy
. A(x,y)^2 + B(x,y)^2 + C(x,y)^2 = 1.
TRIANGLE.
This triangle of coefficients T(n,k) of x^(2*n-2*k+1)*y^(2*k+1)/(2*n+2)! in C(x,y) begins
2;
0, -4;
0, -160, 6;
0, 0, 1344, -8;
0, 0, 145152, -5760, 10;
0, 0, 0, -2534400, 17600, -12;
0, 0, 0, -551755776, 20500480, -43680, 14;
0, 0, 0, 0, 16400384000, -109025280, 94080, -16;
0, 0, 0, 0, 6006289203200, -213971337216, 441423360, -182784, 18;
0, 0, 0, 0, 0, -271368838840320, 1750895247360, -1472507904, 328320, -20;
0, 0, 0, 0, 0, -150055074904670208, 5196968265646080, -10481366827008, 4247147520, -554400, 22;
0, 0, 0, 0, 0, 0, 9574791098375602176, -60514176440205312, 49984638713856, -10935429120, 890560, -24;
0, 0, 0, 0, 0, 0, 7448871207078094438400, -252719117696793313280, 501704844897484800, -200297889792000, 25701561600, -1372800, 26; ...
RELATED FUNCTIONS.
A(x,y) = x + (-1*x^3 - 3*x*y^2)/3! + (1*x^5 - 30*x^3*y^2 + 5*x*y^4)/5! + (-1*x^7 + 315*x^5*y^2 + 525*x^3*y^4 - 7*x*y^6)/7! + (1*x^9 - 1260*x^7*y^2 + 18270*x^5*y^4 - 2940*x^3*y^6 + 9*x*y^8)/9! + (-1*x^11 + 3465*x^9*y^2 - 496650*x^7*y^4 - 695310*x^5*y^6 + 10395*x^3*y^8 - 11*x*y^10)/11! + (1*x^13 - 7722*x^11*y^2 + 4279275*x^9*y^4 - 52144092*x^7*y^6 + 7702695*x^5*y^8 - 28314*x^3*y^10 + 13*x*y^12)/13! + (-1*x^15 + 15015*x^13*y^2 - 22387365*x^11*y^4 + 2701093395*x^9*y^6 + 3472834365*x^7*y^8 - 49252203*x^5*y^10 + 65065*x^3*y^12 - 15*x*y^14)/15! + ...
such that
. A(x,y) = 0 + Integral B(x,y)*C(y,x) - C(x,y)*B(y,x) dx,
. A(y,x) = 0 + Integral B(y,x)*C(x,y) - C(y,x)*B(x,y) dy.
B(x,y) = 1 + (-1*x^2)/2! + (1*x^4)/4! + (-1*x^6 + 120*x^4*y^2)/6! + (1*x^8 - 672*x^6*y^2)/8! + (-1*x^10 + 2160*x^8*y^2 - 120960*x^6*y^4)/10! + (1*x^12 - 5280*x^10*y^2 + 1584000*x^8*y^4)/12! + (-1*x^14 + 10920*x^12*y^2 - 10250240*x^10*y^4 + 482786304*x^8*y^6)/14! + (1*x^16 - 20160*x^14*y^2 + 45427200*x^12*y^4 - 11480268800*x^10*y^6)/16! + ...
such that
. B(x,y) = 1 + Integral C(x,y)*A(y,x) - A(x,y)*C(y,x) dx,
. B(y,x) = 0 + Integral C(y,x)*A(x,y) - A(y,x)*C(x,y) dy.
CC(x) = 1 - 2*x^2/2!^2 + 56*x^4/4!^2 - 8336*x^6/6!^2 + 3985792*x^8/8!^2 - 4679517952*x^10/10!^2 + 11427218287616*x^12/12!^2 - 51793067942397952*x^14/14!^2 + 400951893341645930496*x^16/16!^2 + ... + A326551(n)*x^(2*n)/(2*n)!^2 + ...
such that
. CC(x*y) = sqrt( (Bx*Cy - Cx*By)^2 + (Cx*Ay - Ax*Cy)^2 + (Ax*By - Bx*Ay)^2 ).
SS(x) = x - 8*x^3/3!^2 + 576*x^5/5!^2 - 160768*x^7/7!^2 + 123535360*x^9/9!^2 - 212713734144*x^11/11!^2 + 716196297048064*x^13/13!^2 - 4280584942657732608*x^15/15!^2 + 42250703121584165486592*x^17/17!^2 + ... + A326552(n)*x^(2*n+1)/(2*n+1)!^2 + ...
such that SS(x*y) = Ax*Ay + Bx*By + Cx*Cy.
PROG
(PARI) {TCx(n, k) = my(Ax=x, Bx=1, Cx=x, Ay=y, By=y, Cy=1);
for(i=0, 2*n+1,
Ax = 0 + intformal( Bx*Cy - Cx*By, x) + O(x^(2*n+2));
Bx = 1 + intformal( Cx*Ay - Ax*Cy, x) + O(x^(2*n+2));
Cx = 0 + intformal( Ax*By - Bx*Ay, x) + O(x^(2*n+2));
Ay = 0 + intformal( By*Cx - Cy*Bx, y) + O(y^(2*n+2));
By = 0 + intformal( Cy*Ax - Ay*Cx, y) + O(y^(2*n+2));
Cy = 1 + intformal( Ay*Bx - By*Ax, y) + O(y^(2*n+2));
);
(2*n+2)! * polcoeff( polcoeff(Cx, 2*n-2*k+1, x), 2*k+1, y)}
for(n=0, 10, for(k=0, n, print1( TCx(n, k), ", ")); print(""))
CROSSREFS
Cf. A326797 (A), A326798 (B).
Cf. A326796 (row sums), A326551 (CC), A326552 (SS).
KEYWORD
sign,tabl
AUTHOR
Paul D. Hanna, Aug 03 2019
STATUS
approved
A326801
Consider the e.g.f. C(x,y) = sqrt(1/2) * Sum_{n>=0} Sum_{k=0..2*n} T(n,k) * x^(2*n-k) * y^k / ((2*n-k)!*k!) and related functions S(x,y) and D(x,y), as defined in the Formula section. Sequence gives the triangular array of coefficients T(n,k) (n>=0, 0<=k<=2*n) of C(x,y).
+10
6
1, -1, -1, 0, 1, 0, 0, 1, 0, -1, 0, 8, 8, 0, -1, 0, 1, 0, -24, 0, 0, -24, 0, 1, 0, -1, 0, 48, 0, -576, -576, 0, 48, 0, -1, 0, 1, 0, -80, 0, 3200, 0, 0, 3200, 0, -80, 0, 1, 0, -1, 0, 120, 0, -10240, 0, 160768, 160768, 0, -10240, 0, 120, 0, -1, 0, 1, 0, -168, 0, 24960, 0, -1433600, 0, 0, -1433600, 0, 24960, 0, -168, 0, 1, 0, -1, 0, 224, 0, -51520, 0, 6723584, 0, -123535360, -123535360, 0, 6723584, 0, -51520, 0, 224, 0, -1, 0, 1, 0, -288, 0, 94976, 0, -22586368, 0, 1615675392, 0, 0, 1615675392, 0, -22586368, 0, 94976, 0, -288, 0, 1, 0
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,12
LINKS
Paul D. Hanna, Table of n, a(n) for n = 0..3720 (first 60 rows of this triangle).
FORMULA
The e.g.f. Cx = C(x,y) and related functions Sx = S(x,y), Dx = D(x,y), Sy = S(y,x), Cy = C(y,x), and Dy = D(y,x) satisfy the following relations.
DEFINITION.
(1a) Sx = Integral Cx*Dy + Cy*Dx dx,
(1b) Cx = sqrt(1/2) - Integral Sx*Dy + Sy*Dx dx,
(1c) Dx = sqrt(1/2) - Integral Sx*Cy - Sy*Cx dx,
(2a) Sy = Integral Cy*Dx + Cx*Dy dy,
(2b) Cy = sqrt(1/2) - Integral Sy*Dx + Sx*Dy dy,
(2c) Dy = sqrt(1/2) - Integral Sy*Cx - Sx*Cy dy.
IDENTITIES.
(3a) Dx^2 + Cx^2 + Sx^2 = 1.
(3b) Dy^2 + Cy^2 + Sy^2 = 1.
(4a) Dx*(d/dx Dx) + Cx*(d/dx Cx) + Sx*(d/dx Sx) = 0.
(4b) Dy*(d/dy Dy) + Cy*(d/dy Cy) + Sy*(d/dy Sy) = 0.
(4c) Dy*(d/dx Dx) - Cy*(d/dx Cx) - Sy*(d/dx Sx) = 0.
(4d) Dx*(d/dy Dy) - Cx*(d/dy Cy) - Sx*(d/dy Sy) = 0.
(5a) (Dx*Dy - Cx*Cy - Sx*Sy)^2 + (d/dx Dx)^2 + (d/dx Cx)^2 + (d/dx Sx)^2 = 1.
(5b) (Dx*Dy - Cx*Cy - Sx*Sy)^2 + (d/dy Dy)^2 + (d/dy Cy)^2 + (d/dy Sy)^2 = 1.
RELATED FUNCTIONS.
(6a) SS(x*y) = Dx*Dy - Cx*Cy - Sx*Sy.
(6b) d/dx SS(x*y) = Dx*(d/dx Dy) - Cx*(d/dx Cy) - Sx*(d/dx Sy).
(6c) d/dy SS(x*y) = Dy*(d/dy Dx) - Cy*(d/dy Cx) - Sy*(d/dy Sx).
(7a) CC(x*y)^2 = (Cx*Dy + Cy*Dx)^2 + (Sx*Dy + Sy*Dx)^2 + (Sx*Cy - Sy*Cx)^2.
(7b) CC(x*y)^2 = (d/dx Dx)^2 + (d/dx Cx)^2 + (d/dx Sx)^2.
(7c) CC(x*y)^2 = (d/dy Dy)^2 + (d/dy Cy)^2 + (d/dy Sy)^2.
In the above, CC(x) and SS(x) are the e.g.f.s of A326551 and A326552 defined by
(8a) CC(x*y)^2 + SS(x*y)^2 = 1,
(8b) SS(x*y) = Integral CC(x*y) * (Integral CC(x*y) dy) dx,
(8c) CC(x*y) = 1 - Integral SS(x*y) * (Integral CC(x*y) dy) dx,
(8d) SS(x*y) = sin( Integral Integral CC(x*y) dx dy ),
(8e) CC(x*y) = cos( Integral Integral CC(x*y) dx dy ).
DERIVATIVES.
(9a) d/dx Sx = Cx*Dy + Cy*Dx.
(9b) d/dx Cx = -Sx*Dy - Sy*Dx.
(9c) d/dx Dx = -Sx*Cy + Sy*Cx.
(9d) d/dy Sy = Sy*Dx + Sx*Dy.
(9e) d/dy Cy = -Sy*Dx - Sx*Dy.
(9f) d/dy Dy = -Sy*Cx + Sx*Cy.
EXAMPLE
E.g.f.: C(x,y) = sqrt(1/2) * (1 + (-x^2/2! - x*y ) + ( x^4/4! + x*y^3/3! ) + (-x^6/6! + 8*x^4*y^2/(4!*2!) + 8*x^3*y^3/(3!*3!) - x*y^5/5! ) + ( x^8/8! - 24*x^6*y^2/(6!*2!) - 24*x^3*y^5/(3!*5!) + x*y^7/7! ) + (-x^10/10! + 48*x^8*y^2/(8!*2!) - 576*x^6*y^4/(6!*4!) - 576*x^5*y^5/(5!*5!) + 48*x^3*y^7/(3!*7!) - x*y^9/9! ) + ( x^12/12! - 80*x^10*y^2/(10!*2!) + 3200*x^8*y^4/(8!*4!) + 3200*x^5*y^7/(5!*7!) - 80*x^3*y^9/(3!*9!) + x*y^11/11! ) + (-x^14/14! + 120*x^12*y^2/(12!*2!) - 10240*x^10*y^4/(10!*4!) + 160768*x^8*y^6/(8!*6!) + 160768*x^7*y^7/(7!*7!) - 10240*x^5*y^9/(5!*9!) + 120*x^3*y^11/(3!*11!) - x*y^13/13! ) + ( x^16/16! - 168*x^14*y^2/(14!*2!) + 24960*x^12*y^4/(12!*4!) - 1433600*x^10*y^6/(10!*6!) - 1433600*x^7*y^9/(7!*9!) + 24960*x^5*y^11/(5!*11!) - 168*x^3*y^13/(3!*13!) + x*y^15/15! ) + (-x^18/18! + 224*x^16*y^2/(16!*2!) - 51520*x^14*y^4/(14!*4!) + 6723584*x^12*y^6/(12!*6!) - 123535360*x^10*y^8/(10!*8!) - 123535360*x^9*y^9/(9!*9!) + 6723584*x^7*y^11/(7!*11!) - 51520*x^5*y^13/(5!*13!) + 224*x^3*y^15/(3!*15!) - x*y^17/17! ) + ( x^20/20! - 288*x^18*y^2/(18!*2!) + 94976*x^16*y^4/(16!*4!) - 22586368*x^14*y^6/(14!*6!) + 1615675392*x^12*y^8/(12!*8!) + 1615675392*x^9*y^11/(9!*11!) - 22586368*x^7*y^13/(7!*13!) + 94976*x^5*y^15/(5!*15!) - 288*x^3*y^17/(3!*17!) + x*y^19/19! ) + ...).
This triangle of coefficients T(n,k) of x^(2*n-k)*y^k/((2*n-k)!*k!) in sqrt(2)*C(x,y) begins
1;
-1, -1, 0;
1, 0, 0, 1, 0;
-1, 0, 8, 8, 0, -1, 0;
1, 0, -24, 0, 0, -24, 0, 1, 0;
-1, 0, 48, 0, -576, -576, 0, 48, 0, -1, 0;
1, 0, -80, 0, 3200, 0, 0, 3200, 0, -80, 0, 1, 0;
-1, 0, 120, 0, -10240, 0, 160768, 160768, 0, -10240, 0, 120, 0, -1, 0;
1, 0, -168, 0, 24960, 0, -1433600, 0, 0, -1433600, 0, 24960, 0, -168, 0, 1, 0;
-1, 0, 224, 0, -51520, 0, 6723584, 0, -123535360, -123535360, 0, 6723584, 0, -51520, 0, 224, 0, -1, 0;
1, 0, -288, 0, 94976, 0, -22586368, 0, 1615675392, 0, 0, 1615675392, 0, -22586368, 0, 94976, 0, -288, 0, 1, 0;
-1, 0, 360, 0, -161280, 0, 61458432, 0, -10447847424, 0, 212713734144, 212713734144, 0, -10447847424, 0, 61458432, 0, -161280, 0, 360, 0, -1, 0;
1, 0, -440, 0, 257280, 0, -144420864, 0, 46282211328, 0, -3835832827904, 0, 0, -3835832827904, 0, 46282211328, 0, -144420864, 0, 257280, 0, -440, 0, 1, 0; ...
CENTRAL TERMS.
The central terms are found in 1 - SS(x*y) = 1 - Dx*Dy + Cx*Cy + Sx*Sy:
[1, -1, 0, 8, 0, -576, 0, 160768, 0, -123535360, 0, 212713734144, 0, -716196297048064, 0, 4280584942657732608, ...] (cf. A326552).
RELATED SERIES.
The e.g.f. of A326800 begins
S(x,y) = x + (-x^3/3! - x*y^2/2! ) + ( x^5/5! - 3*x^3*y^2/(3!*2!) + x*y^4/4! ) + (-x^7/7! + 15*x^5*y^2/(5!*2!) + 15*x^3*y^4/(3!*4!) - x*y^6/6! ) + ( x^9/9! - 35*x^7*y^2/(7!*2!) + 145*x^5*y^4/(5!*4!) - 35*x^3*y^6/(3!*6!) + x*y^8/8! ) + (-x^11/11! + 63*x^9*y^2/(9!*2!) - 1505*x^7*y^4/(7!*4!) - 1505*x^5*y^6/(5!*6!) + 63*x^3*y^8/(3!*8!) - x*y^10/10! ) + ( x^13/13! - 99*x^11*y^2/(11!*2!) + 5985*x^9*y^4/(9!*4!) - 30387*x^7*y^6/(7!*6!) + 5985*x^5*y^8/(5!*8!) - 99*x^3*y^10/(3!*10!) + x*y^12/12! ) + (-x^15/15! + 143*x^13*y^2/(13!*2!) - 16401*x^11*y^4/(11!*4!) + 539679*x^9*y^6/(9!*6!) + 539679*x^7*y^8/(7!*8!) - 16401*x^5*y^10/(5!*10!) + 143*x^3*y^12/(3!*12!) - x*y^14/14! ) + ...
The e.g.f. of A326802 begins
D(x,y) = sqrt(1/2) * (1 + (-x^2/2! + x*y ) + ( x^4/4! - x*y^3/3! ) + (-x^6/6! + 8*x^4*y^2/(4!*2!) - 8*x^3*y^3/(3!*3!) + x*y^5/5! ) + ( x^8/8! - 24*x^6*y^2/(6!*2!) + 24*x^3*y^5/(3!*5!) - x*y^7/7! ) + (-x^10/10! + 48*x^8*y^2/(8!*2!) - 576*x^6*y^4/(6!*4!) + 576*x^5*y^5/(5!*5!) - 48*x^3*y^7/(3!*7!) + x*y^9/9! ) + ( x^12/12! - 80*x^10*y^2/(10!*2!) + 3200*x^8*y^4/(8!*4!) - 3200*x^5*y^7/(5!*7!) + 80*x^3*y^9/(3!*9!) - x*y^11/11! ) + (-x^14/14! + 120*x^12*y^2/(12!*2!) - 10240*x^10*y^4/(10!*4!) + 160768*x^8*y^6/(8!*6!) - 160768*x^7*y^7/(7!*7!) + 10240*x^5*y^9/(5!*9!) - 120*x^3*y^11/(3!*11!) + x*y^13/13! ) + ...).
The e.g.f. of A326552 begins
SS(x*y) = (x*y) - 8*(x*y)^3/3!^2 + 576*(x*y)^5/5!^2 - 160768*(x*y)^7/7!^2 + 123535360*(x*y)^9/9!^2 - 212713734144*(x*y)^11/11!^2 + 716196297048064*(x*y)^13/13!^2 - 4280584942657732608*(x*y)^15/15!^2 + 42250703121584165486592*(x*y)^17/17!^2 - 651154631135458759089848320*(x*y)^19/19!^2 + 14983590319172065236171175755776*(x*y)^21/21!^2 + ... + A326552(n)*(x*y)^(2*n-1)/(2*n-1)! + ...
such that
SS(x*y) = Dx*Dy - Cx*Cy - Sx*Sy.
The e.g.f. of A326551 begins
CC(x*y) = 1 - 2*(x*y)^2/2!^2 + 56*(x*y)^4/4!^2 - 8336*(x*y)^6/6!^2 + 3985792*(x*y)^8/8!^2 - 4679517952*(x*y)^10/10!^2 + 11427218287616*(x*y)^12/12!^2 - 51793067942397952*(x*y)^14/14!^2 + 400951893341645930496*(x*y)^16/16!^2 - 4975999084909976839454720*(x*y)^18/18!^2 + 94178912073481319162642169856*(x*y)^20/20!^2 -+ ... + A326551(n)*(x*y)^(2*n)/(2*n)! + ...
such that
CC(x*y)^2 = (Cx*Dy + Cy*Dx)^2 + (Sx*Dy + Sy*Dx)^2 + (Sx*Cy - Sy*Cx)^2,
and CC(x*Y)^2 + SS(x*y)^2 = 1.
PROG
(PARI)
{TCx(n, k) = my(Cx=1, Sx=x, Dx=1, Cy=1, Sy=y, Dy=1);
for(i=0, 2*n+1,
Sx = intformal( Cx*Dy + Cy*Dx, x) + O(x^(2*n+2));
Cx = sqrt(1/2) - intformal( Sx*Dy + Sy*Dx, x);
Dx = sqrt(1/2) - intformal( Sx*Cy - Sy*Cx, x);
Sy = intformal( Cy*Dx + Cx*Dy, y) + O(y^(2*n+2));
Cy = sqrt(1/2) - intformal( Sy*Dx + Sx*Dy, y);
Dy = sqrt(1/2) - intformal( Sy*Cx - Sx*Cy, y);
);
round( (2*n-k)!*k! * polcoeff( polcoeff(sqrt(2)*Cx, 2*n-k, x), k, y) )}
for(n=0, 10, for(k=0, 2*n, print1( TCx(n, k), ", ")); print(""))
CROSSREFS
Cf. A326800 (Sx), A326802 (Dx), A326551 (CC), A326552 (SS).
KEYWORD
sign,tabf
AUTHOR
Paul D. Hanna, Jul 27 2019
STATUS
approved
A326802
Consider the e.g.f. D(x,y) = sqrt(1/2) * Sum_{n>=0} Sum_{k=0..2*n} T(n,k) * x^(2*n-k) * y^k / ((2*n-k)!*k!) and related functions S(x,y) and C(x,y), as defined in the Formula section. Sequence gives the triangular array of coefficients T(n,k) (n>=0, 0<=k<=2*n) of D(x,y).
+10
6
1, -1, 1, 0, 1, 0, 0, -1, 0, -1, 0, 8, -8, 0, 1, 0, 1, 0, -24, 0, 0, 24, 0, -1, 0, -1, 0, 48, 0, -576, 576, 0, -48, 0, 1, 0, 1, 0, -80, 0, 3200, 0, 0, -3200, 0, 80, 0, -1, 0, -1, 0, 120, 0, -10240, 0, 160768, -160768, 0, 10240, 0, -120, 0, 1, 0, 1, 0, -168, 0, 24960, 0, -1433600, 0, 0, 1433600, 0, -24960, 0, 168, 0, -1, 0, -1, 0, 224, 0, -51520, 0, 6723584, 0, -123535360, 123535360, 0, -6723584, 0, 51520, 0, -224, 0, 1, 0, 1, 0, -288, 0, 94976, 0, -22586368, 0, 1615675392, 0, 0, -1615675392, 0, 22586368, 0, -94976, 0, 288, 0, -1, 0
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,12
LINKS
Paul D. Hanna, Table of n, a(n) for n = 0..3720 (the first 60 rows of the triangle).
FORMULA
The e.g.f. Dx = D(x,y) and related functions Sx = S(x,y), Cx = C(x,y), Sy = S(y,x), Cy = C(y,x), and Dy = D(y,x) satisfy the following relations.
DEFINITION.
(1a) Sx = Integral Cx*Dy + Cy*Dx dx,
(1b) Cx = sqrt(1/2) - Integral Sx*Dy + Sy*Dx dx,
(1c) Dx = sqrt(1/2) - Integral Sx*Cy - Sy*Cx dx,
(2a) Sy = Integral Cy*Dx + Cx*Dy dy,
(2b) Cy = sqrt(1/2) - Integral Sy*Dx + Sx*Dy dy,
(2c) Dy = sqrt(1/2) - Integral Sy*Cx - Sx*Cy dy.
IDENTITIES.
(3a) Dx^2 + Cx^2 + Sx^2 = 1.
(3b) Dy^2 + Cy^2 + Sy^2 = 1.
(4a) Dx*(d/dx Dx) + Cx*(d/dx Cx) + Sx*(d/dx Sx) = 0.
(4b) Dy*(d/dy Dy) + Cy*(d/dy Cy) + Sy*(d/dy Sy) = 0.
(4c) Dy*(d/dx Dx) - Cy*(d/dx Cx) - Sy*(d/dx Sx) = 0.
(4d) Dx*(d/dy Dy) - Cx*(d/dy Cy) - Sx*(d/dy Sy) = 0.
(5a) (Dx*Dy - Cx*Cy - Sx*Sy)^2 + (d/dx Dx)^2 + (d/dx Cx)^2 + (d/dx Sx)^2 = 1.
(5b) (Dx*Dy - Cx*Cy - Sx*Sy)^2 + (d/dy Dy)^2 + (d/dy Cy)^2 + (d/dy Sy)^2 = 1.
RELATED FUNCTIONS.
(6a) SS(x*y) = Dx*Dy - Cx*Cy - Sx*Sy.
(6b) d/dx SS(x*y) = Dx*(d/dx Dy) - Cx*(d/dx Cy) - Sx*(d/dx Sy).
(6c) d/dy SS(x*y) = Dy*(d/dy Dx) - Cy*(d/dy Cx) - Sy*(d/dy Sx).
(7a) CC(x*y)^2 = (Cx*Dy + Cy*Dx)^2 + (Sx*Dy + Sy*Dx)^2 + (Sx*Cy - Sy*Cx)^2.
(7b) CC(x*y)^2 = (d/dx Dx)^2 + (d/dx Cx)^2 + (d/dx Sx)^2.
(7c) CC(x*y)^2 = (d/dy Dy)^2 + (d/dy Cy)^2 + (d/dy Sy)^2.
In the above, CC(x) and SS(x) are the e.g.f.s of A326551 and A326552 defined by
(8a) CC(x*y)^2 + SS(x*y)^2 = 1,
(8b) SS(x*y) = Integral CC(x*y) * (Integral CC(x*y) dy) dx,
(8c) CC(x*y) = 1 - Integral SS(x*y) * (Integral CC(x*y) dy) dx,
(8d) SS(x*y) = sin( Integral Integral CC(x*y) dx dy ),
(8e) CC(x*y) = cos( Integral Integral CC(x*y) dx dy ).
DERIVATIVES.
(9a) d/dx Sx = Cx*Dy + Cy*Dx.
(9b) d/dx Cx = -Sx*Dy - Sy*Dx.
(9c) d/dx Dx = -Sx*Cy + Sy*Cx.
(9d) d/dy Sy = Sy*Dx + Sx*Dy.
(9e) d/dy Cy = -Sy*Dx - Sx*Dy.
(9f) d/dy Dy = -Sy*Cx + Sx*Cy.
EXAMPLE
E.g.f.: D(x,y) = sqrt(1/2) * (1 + (-x^2/2! + x*y ) + ( x^4/4! - x*y^3/3! ) + (-x^6/6! + 8*x^4*y^2/(4!*2!) - 8*x^3*y^3/(3!*3!) + x*y^5/5! ) + ( x^8/8! - 24*x^6*y^2/(6!*2!) + 24*x^3*y^5/(3!*5!) - x*y^7/7! ) + (-x^10/10! + 48*x^8*y^2/(8!*2!) - 576*x^6*y^4/(6!*4!) + 576*x^5*y^5/(5!*5!) - 48*x^3*y^7/(3!*7!) + x*y^9/9! ) + ( x^12/12! - 80*x^10*y^2/(10!*2!) + 3200*x^8*y^4/(8!*4!) - 3200*x^5*y^7/(5!*7!) + 80*x^3*y^9/(3!*9!) - x*y^11/11! ) + (-x^14/14! + 120*x^12*y^2/(12!*2!) - 10240*x^10*y^4/(10!*4!) + 160768*x^8*y^6/(8!*6!) - 160768*x^7*y^7/(7!*7!) + 10240*x^5*y^9/(5!*9!) - 120*x^3*y^11/(3!*11!) + x*y^13/13! ) + ( x^16/16! - 168*x^14*y^2/(14!*2!) + 24960*x^12*y^4/(12!*4!) - 1433600*x^10*y^6/(10!*6!) + 1433600*x^7*y^9/(7!*9!) - 24960*x^5*y^11/(5!*11!) + 168*x^3*y^13/(3!*13!) - x*y^15/15! ) + (-1*x^18/18! + 224*x^16*y^2/(16!*2!) - 51520*x^14*y^4/(14!*4!) + 6723584*x^12*y^6/(12!*6!) - 123535360*x^10*y^8/(10!*8!) + 123535360*x^9*y^9/(9!*9!) - 6723584*x^7*y^11/(7!*11!) + 51520*x^5*y^13/(5!*13!) - 224*x^3*y^15/(3!*15!) + x*y^17/17! ) + ( x^20/20! - 288*x^18*y^2/(18!*2!) + 94976*x^16*y^4/(16!*4!) - 22586368*x^14*y^6/(14!*6!) + 1615675392*x^12*y^8/(12!*8!) - 1615675392*x^9*y^11/(9!*11!) + 22586368*x^7*y^13/(7!*13!) - 94976*x^5*y^15/(5!*15!) + 288*x^3*y^17/(3!*17!) - x*y^19/19! ) + ...).
This triangle of coefficients T(n,k) of x^(2*n-k)*y^k/((2*n-k)!*k!) in sqrt(2)*D(x,y) begins
1;
-1, 1, 0;
1, 0, 0, -1, 0;
-1, 0, 8, -8, 0, 1, 0;
1, 0, -24, 0, 0, 24, 0, -1, 0;
-1, 0, 48, 0, -576, 576, 0, -48, 0, 1, 0;
1, 0, -80, 0, 3200, 0, 0, -3200, 0, 80, 0, -1, 0;
-1, 0, 120, 0, -10240, 0, 160768, -160768, 0, 10240, 0, -120, 0, 1, 0;
1, 0, -168, 0, 24960, 0, -1433600, 0, 0, 1433600, 0, -24960, 0, 168, 0, -1, 0;
-1, 0, 224, 0, -51520, 0, 6723584, 0, -123535360, 123535360, 0, -6723584, 0, 51520, 0, -224, 0, 1, 0;
1, 0, -288, 0, 94976, 0, -22586368, 0, 1615675392, 0, 0, -1615675392, 0, 22586368, 0, -94976, 0, 288, 0, -1, 0;
-1, 0, 360, 0, -161280, 0, 61458432, 0, -10447847424, 0, 212713734144, -212713734144, 0, 10447847424, 0, -61458432, 0, 161280, 0, -360, 0, 1, 0;
1, 0, -440, 0, 257280, 0, -144420864, 0, 46282211328, 0, -3835832827904, 0, 0, 3835832827904, 0, -46282211328, 0, 144420864, 0, -257280, 0, 440, 0, -1, 0; ...
CENTRAL TERMS.
The central terms are found in 1 + SS(x*y) = 1 + Dx*Dy - Cx*Cy - Sx*Sy:
[1, 1, 0, -8, 0, 576, 0, -160768, 0, 123535360, 0, -212713734144, 0, 716196297048064, 0, -4280584942657732608, ...] (cf. A326552).
RELATED SERIES.
The e.g.f. of A326800 begins
S(x,y) = x + (-x^3/3! - x*y^2/2! ) + ( x^5/5! - 3*x^3*y^2/(3!*2!) + x*y^4/4! ) + (-x^7/7! + 15*x^5*y^2/(5!*2!) + 15*x^3*y^4/(3!*4!) - x*y^6/6! ) + ( x^9/9! - 35*x^7*y^2/(7!*2!) + 145*x^5*y^4/(5!*4!) - 35*x^3*y^6/(3!*6!) + x*y^8/8! ) + (-x^11/11! + 63*x^9*y^2/(9!*2!) - 1505*x^7*y^4/(7!*4!) - 1505*x^5*y^6/(5!*6!) + 63*x^3*y^8/(3!*8!) - x*y^10/10! ) + ( x^13/13! - 99*x^11*y^2/(11!*2!) + 5985*x^9*y^4/(9!*4!) - 30387*x^7*y^6/(7!*6!) + 5985*x^5*y^8/(5!*8!) - 99*x^3*y^10/(3!*10!) + x*y^12/12! ) + (-x^15/15! + 143*x^13*y^2/(13!*2!) - 16401*x^11*y^4/(11!*4!) + 539679*x^9*y^6/(9!*6!) + 539679*x^7*y^8/(7!*8!) - 16401*x^5*y^10/(5!*10!) + 143*x^3*y^12/(3!*12!) - x*y^14/14! ) + ...
The e.g.f. of A326801 begins
C(x,y) = sqrt(1/2) * (1 + (-x^2/2! - x*y ) + ( x^4/4! + x*y^3/3! ) + (-x^6/6! + 8*x^4*y^2/(4!*2!) + 8*x^3*y^3/(3!*3!) - x*y^5/5! ) + ( x^8/8! - 24*x^6*y^2/(6!*2!) - 24*x^3*y^5/(3!*5!) + x*y^7/7! ) + (-x^10/10! + 48*x^8*y^2/(8!*2!) - 576*x^6*y^4/(6!*4!) - 576*x^5*y^5/(5!*5!) + 48*x^3*y^7/(3!*7!) - x*y^9/9! ) + ( x^12/12! - 80*x^10*y^2/(10!*2!) + 3200*x^8*y^4/(8!*4!) + 3200*x^5*y^7/(5!*7!) - 80*x^3*y^9/(3!*9!) + x*y^11/11! ) + (-x^14/14! + 120*x^12*y^2/(12!*2!) - 10240*x^10*y^4/(10!*4!) + 160768*x^8*y^6/(8!*6!) + 160768*x^7*y^7/(7!*7!) - 10240*x^5*y^9/(5!*9!) + 120*x^3*y^11/(3!*11!) - x*y^13/13! ) + ...).
The e.g.f. of A326552 begins
SS(x*y) = (x*y) - 8*(x*y)^3/3!^2 + 576*(x*y)^5/5!^2 - 160768*(x*y)^7/7!^2 + 123535360*(x*y)^9/9!^2 - 212713734144*(x*y)^11/11!^2 + 716196297048064*(x*y)^13/13!^2 - 4280584942657732608*(x*y)^15/15!^2 + 42250703121584165486592*(x*y)^17/17!^2 - 651154631135458759089848320*(x*y)^19/19!^2 + 14983590319172065236171175755776*(x*y)^21/21!^2 + ... + A326552(n)*(x*y)^(2*n-1)/(2*n-1)! + ...
such that
SS(x*y) = Dx*Dy - Cx*Cy - Sx*Sy.
The e.g.f. of A326551 begins
CC(x*y) = 1 - 2*(x*y)^2/2!^2 + 56*(x*y)^4/4!^2 - 8336*(x*y)^6/6!^2 + 3985792*(x*y)^8/8!^2 - 4679517952*(x*y)^10/10!^2 + 11427218287616*(x*y)^12/12!^2 - 51793067942397952*(x*y)^14/14!^2 + 400951893341645930496*(x*y)^16/16!^2 - 4975999084909976839454720*(x*y)^18/18!^2 + 94178912073481319162642169856*(x*y)^20/20!^2 -+ ... + A326551(n)*(x*y)^(2*n)/(2*n)! + ...
such that
CC(x*y)^2 = (Cx*Dy + Cy*Dx)^2 + (Sx*Dy + Sy*Dx)^2 + (Sx*Cy - Sy*Cx)^2,
and CC(x*Y)^2 + SS(x*y)^2 = 1.
PROG
(PARI)
{TDx(n, k) = my(Cx=1, Sx=x, Dx=1, Cy=1, Sy=y, Dy=1);
for(i=0, 2*n+1,
Sx = intformal( Cx*Dy + Cy*Dx, x) + O(x^(2*n+2));
Cx = sqrt(1/2) - intformal( Sx*Dy + Sy*Dx, x);
Dx = sqrt(1/2) - intformal( Sx*Cy - Sy*Cx, x);
Sy = intformal( Cy*Dx + Cx*Dy, y) + O(y^(2*n+2));
Cy = sqrt(1/2) - intformal( Sy*Dx + Sx*Dy, y);
Dy = sqrt(1/2) - intformal( Sy*Cx - Sx*Cy, y);
);
round( (2*n-k)!*k! * polcoeff( polcoeff(sqrt(2)*Dx, 2*n-k, x), k, y) )}
for(n=0, 10, for(k=0, 2*n, print1( TDx(n, k), ", ")); print(""))
CROSSREFS
Cf. A326800 (Sx), A326801 (Cx), A326551 (CC), A326552 (SS).
KEYWORD
sign,tabf
AUTHOR
Paul D. Hanna, Jul 27 2019
STATUS
approved
A325291
E.g.f. C(x), where C(x*y) + S(x*y) = exp( Integral Integral C(x*y) dx dy ) such that C(x)^2 - S(x)^2 = 1.
+10
4
1, 2, 56, 8336, 3985792, 4679517952, 11427218287616, 51793067942397952, 400951893341645930496, 4975999084909976839454720, 94178912073481319162642169856, 2610878440961060713599511173791744, 102545703927828194073741484514193965056, 5548919569628098800740786379865766154469376, 403949193167852851803947801218003477783686152192
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
COMMENTS
Unsigned version of A326551.
LINKS
Paul D. Hanna, Table of n, a(n) for n = 0..200
FORMULA
E.g.f. C(x) = Sum_{n>=0} a(n)*x^(2*n)/(2*n)!^2, where series C(x) and related series S(x) satisfy the following relations.
(1.a) C(x)^2 - S(x)^2 = 1.
(1.b) C'(x)/S(x) = S'(x)/C(x) = 1/x * Integral C(x) dx.
(2.a) S(x) = Integral C(x)/x * (Integral C(x) dx) dx.
(2.b) C(x) = 1 + Integral S(x)/x * (Integral C(x) dx) dx.
(3.a) C(x) + S(x) = exp( Integral 1/x * (Integral C(x) dx) dx ).
(3.b) C(x) = cosh( Integral 1/x * (Integral C(x) dx) dx ).
(3.c) S(x) = sinh( Integral 1/x * (Integral C(x) dx) dx ).
Integration.
(4.a) S(x*y) = Integral C(x*y) * (Integral C(x*y) dy) dx.
(4.b) C(x*y) = 1 + Integral S(x*y) * (Integral C(x*y) dy) dx.
(4.c) S(x*y) = Integral C(x*y) * (Integral C(x*y) dx) dy.
(4.d) C(x*y) = 1 + Integral S(x*y) * (Integral C(x*y) dx) dy.
Exponential.
(5.a) C(x*y) + S(x*y) = exp( Integral Integral C(x*y) dx dy ).
(5.b) C(x*y) = cosh( Integral Integral C(x*y) dx dy ).
(5.c) S(x*y) = sinh( Integral Integral C(x*y) dx dy ).
Derivatives.
(6.a) d/dx S(x*y) = C(x*y) * Integral C(x*y) dy.
(6.b) d/dx C(x*y) = S(x*y) * Integral C(x*y) dy.
(6.c) d/dy S(x*y) = C(x*y) * Integral C(x*y) dx.
(6.d) d/dy C(x*y) = S(x*y) * Integral C(x*y) dx.
EXAMPLE
E.g.f. C(x) = 1 + 2*x^2/2!^2 + 56*x^4/4!^2 + 8336*x^6/6!^2 + 3985792*x^8/8!^2 + 4679517952*x^10/10!^2 + 11427218287616*x^12/12!^2 + 51793067942397952*x^14/14!^2 + 400951893341645930496*x^16/16!^2 + 4975999084909976839454720*x^18/18!^2 + 94178912073481319162642169856*x^20/20!^2 + ...
where C(x) = cosh( Integral 1/x * (Integral C(x) dx) dx ),
also, C(x*y) = cosh( Integral Integral C(x*y) dx dy ).
RELATED SERIES.
S(x) = x + 8*x^3/3!^2 + 576*x^5/5!^2 + 160768*x^7/7!^2 + 123535360*x^9/9!^2 + 212713734144*x^11/11!^2 + 716196297048064*x^13/13!^2 + 4280584942657732608*x^15/15!^2 + 42250703121584165486592*x^17/17!^2 + 651154631135458759089848320*x^19/19!^2 + 14983590319172065236171175755776*x^21/21!^2 + ...
where S(x) = sinh( Integral 1/x * (Integral C(x) dx) dx ),
also, S(x*y) = sinh( Integral Integral C(x*y) dx dy ).
SPECIFIC VALUES.
At x = 1/2,
C(1/2) = 1.13133757946411922642102833324416139...
S(1/2) = 0.52907912329606456055608764850290077...
log(C(1/2) + S(1/2)) = 0.50706859662590456104854330721421537...
At x = 1,
C(1) = 1.61616724447561044622618032294959193...
S(1) = 1.26964426597212165112687564431552303...
log(C(1) + S(1)) = 1.05980614652360497313310791544203867...
At x = 2,
C(2) = 7.0181980831554020705059330009720760...
S(2) = 6.9465894030384550946994132182413166...
log(C(2) + S(2)) = 2.636538981679765615420983831302958...
At x = 3, the power series for C(x) and S(x) diverge.
PROG
(PARI) {a(n) = my(C=1, S=x); for(i=1, 2*n,
S = intformal( C/x * intformal( C +x*O(x^(2*n)) ) );
C = 1 + intformal( S/x * intformal( C +x*O(x^(2*n)) ) ); ); (2*n)!^2*polcoeff(C, 2*n)}
for(n=0, 30, print1(a(n), ", "))
CROSSREFS
Cf. A325290 (C+S), A325292 (S).
Cf. A326551.
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Apr 16 2019
STATUS
approved
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