Search: a326801 -id:a326801
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A326551
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E.g.f. C(x), where C(x*y) + i*S(x*y) = exp( i*Integral Integral C(x*y) dx dy ) such that C(x)^2 + S(x)^2 = 1.
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+10
9
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1, -2, 56, -8336, 3985792, -4679517952, 11427218287616, -51793067942397952, 400951893341645930496, -4975999084909976839454720, 94178912073481319162642169856, -2610878440961060713599511173791744, 102545703927828194073741484514193965056, -5548919569628098800740786379865766154469376, 403949193167852851803947801218003477783686152192
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OFFSET
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0,2
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COMMENTS
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The hyperbolic analog of the e.g.f. is described by A325291.
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LINKS
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FORMULA
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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) 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.
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EXAMPLE
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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) = cos( Integral 1/x * (Integral C(x) dx) dx ),
also, C(x*y) = cos( 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) = 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.
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PROG
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(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), ", "))
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CROSSREFS
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KEYWORD
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sign
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AUTHOR
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STATUS
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approved
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A326552
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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.
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+10
8
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1, -8, 576, -160768, 123535360, -212713734144, 716196297048064, -4280584942657732608, 42250703121584165486592, -651154631135458759089848320, 14983590319172065236171175755776, -496301942561421311900528265903734784, 22953613919171561374366988621726483480576, -1444609513446024762131466039751756562435145728, 121022534222796916421149671221445519229890299166720
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OFFSET
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1,2
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COMMENTS
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The hyperbolic analog of the e.g.f. is described by A325292.
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LINKS
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FORMULA
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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.
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EXAMPLE
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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! + ...
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PROG
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(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), ", "))
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CROSSREFS
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KEYWORD
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sign
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AUTHOR
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STATUS
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approved
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A326800
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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).
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+10
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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
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OFFSET
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1,5
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COMMENTS
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The e.g.f. S(x,y) is equivalent to the e.g.f. of A326797.
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LINKS
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FORMULA
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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.
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EXAMPLE
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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.
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! ) + ...).
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! ) + ...).
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.
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.
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PROG
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(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(""))
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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A326802
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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).
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+10
6
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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)
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OFFSET
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0,12
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LINKS
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FORMULA
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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.
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EXAMPLE
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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.
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! ) + ...
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! ) + ...).
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.
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.
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PROG
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(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(""))
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CROSSREFS
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KEYWORD
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sign,tabf
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STATUS
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approved
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A326556
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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.
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+10
1
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1, -4, 256, -67072, 49479680, -82817122304, 273099601739776, -1606512897507196928, 15659025634284911198208, -238894370882781809622384640, 5451274531297360096585324691456, -179296966081016547805899589056200704, 8242844472527700570663352676068232265728, -516102091343047279882754030489835708929277952, 43042816831864259208854418353099287467922680709120
(list;
graph;
refs;
listen;
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text;
internal format)
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OFFSET
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0,2
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COMMENTS
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LINKS
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EXAMPLE
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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 ).
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PROG
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(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), ", "))
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CROSSREFS
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KEYWORD
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sign
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AUTHOR
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STATUS
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approved
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