[go: up one dir, main page]
login
The OEIS is supported by the many generous donors to the OEIS Foundation.
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
Search: a143601 -id:a143601
Displaying 1-10 of 11 results found.
page 1 2
     Sort: relevance | references | number | modified | created      Format: long | short | data
A143598
E.g.f.: A(x) = exp(x*sinh(x*G(x))) where G(x) = cosh(x*G(x)) is the e.g.f. of A143601.
+20
0
1, 2, 28, 1176, 103440, 15726880, 3684098496, 1232799974784, 558670427013376, 329559835063067136, 245462725323910487040, 225319148634038399801344, 249936012383478860884217856, 329609037187846742271984869376
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
LINKS
Table of n, a(n) for n=0..13.
FORMULA
E.g.f.: A(x) = exp(x*F(x)) where F(x) is the e.g.f. of A007106.
E.g.f.: A(x) = sqrt(H(x)*H(-x)) where H(x) = exp(x*sqrt(H(x)/H(-x))) is the e.g.f. of A143599.
E.g.f. satisfies: A(x/cosh(x)) = exp(x*tanh(x)). [From Paul D. Hanna, Aug 29 2008]
EXAMPLE
E.g.f.: A(x) = 1 + 2*x^2/2! + 28*x^4/4! + 1176*x^6/6! + 103440*x^8/8! +...
A(x) = exp(x*F(x)) where F(x) = e.g.f. of A007106:
F(x) = x + 4*x^3/3! + 96*x^5/5! + 5888*x^7/7! + 686080*x^9/9! +...
A(x) = exp(x*sqrt(G(x)^2 - 1)) where G(x) = e.g.f. of A143601:
G(x) = 1 + x^2/2! + 13*x^4/4! + 541*x^6/6! + 47545*x^8/8! +...
A(x) = sqrt(H(x)*H(-x)) where H(x) = e.g.f. of A143599:
H(x) = 1 + x + 3*x^2/2! + 10*x^3/3! + 53*x^4/4! + 316*x^5/5! +...
PROG
(PARI) {a(n)=local(G=1+x*O(x^n)); for(i=0, n, G=cosh(x*G)); n!*polcoeff(exp(x*sqrt(G^2-1)), n)}
CROSSREFS
Cf. A058014, A143600, A143601, A007106.
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Aug 27 2008
STATUS
approved
A007106
Number of labeled odd degree trees with 2n nodes.
(Formerly M3704)
+10
15
1, 4, 96, 5888, 686080, 130179072, 36590059520, 14290429935616, 7405376630685696, 4917457306800619520, 4071967909087792857088, 4113850542422629363482624, 4980673081258443273955966976, 7119048451600750435732824260608, 11861520124846917915630931846103040
(list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
REFERENCES
R. W. Robinson, personal communication.
R. W. Robinson, Numerical implementation of graph counting algorithms, AGRC Grant, Math. Dept., Univ. Newcastle, Australia, 1976.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
Seiichi Manyama, Table of n, a(n) for n = 1..211 (terms 1..39 from R. W. Robinson)
Alexander Burstein and Louis W. Shapiro, Pseudo-involutions in the Riordan group, arXiv:2112.11595 [math.CO], 2021.
B. R. Jones, On tree hook length formulas, Feynman rules and B-series, Master's thesis, Simon Fraser University, 2014.
Mathematics Stack Exchange, Marko R. Riedel, Odd degree trees
Mathematics Stack Exchange, Marko R. Riedel, Odd degree trees II
Marko Riedel, Count by Prüfer codes and Stirling numbers
Index entries for sequences related to trees
FORMULA
a(n) = A060279(n)/(2*n). - Vladeta Jovovic, Feb 08 2005
Bisection of A058014. The expansion 1/sqrt(1+x^2)*arcsinh(x) = x - 4*x^3/3! + 64*x^5/5! - ... (see A002454) has series reversion x + 4*x^3/3! + 96*x^5/5! + 5888*x^7/7! + .... The coefficients appear to be the terms of this sequence. As an x-adic limit this e.g.f. equals lim_{n -> infinity} sinh(f(n,x)), where f(0,x) = x and f(n,x) = x*cosh(f(n-1,x)) for n >= 1. See the example section below. - Peter Bala, Apr 24 2012
a(n) = Sum_{k=1..n} binomial(n,k) * k! * (n-2)! [z^{n-2}] [u^k] exp(u(exp(z)+exp(-z)-2)/2)). - Marko Riedel, Jun 16 2016
From Alexander Burstein, Oct 13 2021: (Start)
a(n) = (1/2) * Sum_{k=0..n-1} binomial(2*n,k) * (n-k)^(2*n-2) for n >= 2.
a(n) = (2*n-1)!*[x^(2*n-1)] sinh(REVERT(x/cosh(x))), see A036778. (End)
a(n) = Sum_{k=0..n-1} A156289(n-1, k)*(2*n)!/(2*n-k)!. - Peter Luschny, May 07 2022
EXAMPLE
From Peter Bala, Apr 24 2012: (Start)
Let G(x) = 1 + x^2/2! + 13*x^4/4! + 541*x^6/6! + ... be the e.g.f. for A143601. Then sinh(x*G(x)) = x + 4*x^3/3! + 96*x^5/5! + 5888*x^7/7! + ....
Conjectural e.g.f. as an x-adic limit:
sinh(x) = x + ...; sinh(x*cosh(x)) = x + 4*x^3/3! + ...;
sinh(x*cosh(x*cosh(x))) = x + 4*x^3/3! + 96*x^5/5! + ...;
sinh(x*cosh(x*cosh(x*cosh(x)))) = x + 4*x^3/3! + 96*x^5/5! + 5888*x^7/7! + ....
(End)
MAPLE
A007106(n) = A(2n) where n>=2, A(n) = (add(binomial(n, q)*(n-2*q)^(n-2)/(n-2)!, q=0..n) - add(binomial(n-1, q)*(n-2*q)^(n-3)/(n-3)!, q=0..n-1) + add(binomial(n-1, q)*(n-2-2*q)^(n-3)/(n-3)!, q=0..n-1))*n!/2^(n+1)/(n-1)
MATHEMATICA
{1}~Join~Array[(1/2)*Sum[Binomial[2 #, k]*(# - k)^(2 # - 2), {k, 0, # - 1}] &, 12, 2] (* Michael De Vlieger, Oct 13 2021 *)
PROG
(PARI) a(n) = if(n<=1, n==1, sum(k=0, n-1, binomial(2*n, k) * (n-k)^(2*n-2))/2) \\ Andrew Howroyd, Nov 22 2021
CROSSREFS
Cf. A036778, A058014, A060279, A143601, A156289.
KEYWORD
nonn
AUTHOR
N. J. A. Sloane
EXTENSIONS
Corrected and extended by Vladeta Jovovic, Feb 08 2005
STATUS
approved
A058014
Number of labeled trees with n+1 nodes such that the degrees of all nodes, excluding the first node, are odd.
+10
10
1, 1, 1, 4, 13, 96, 541, 5888, 47545, 686080, 7231801, 130179072, 1695106117, 36590059520, 567547087381, 14290429935616, 257320926233329, 7405376630685696, 151856004814953841, 4917457306800619520
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,4
LINKS
Seiichi Manyama, Table of n, a(n) for n = 0..422
Alexander Postnikov, Papers.
A. Postnikov and R. P. Stanley, Deformations of Coxeter hyperplane arrangements, J. Combin. Theory, Ser. A, 91 (2000), 544-597. (Section 10.2.)
FORMULA
a(n) = (1/2^n) * Sum_{k=0..n} binomial(n,k) * (n + 1 - 2*k)^(n-1).
From Paul D. Hanna, Mar 29 2008: (Start)
E.g.f. satisfies A(x) = exp( x*[A(x) + 1/A(x)]/2 ).
E.g.f. A(x) equals the inverse function of 2*x*log(x)/(1 + x^2).
Let r = radius of convergence of A(x), then r = 0.6627434193491815809747420971092529070562335491150224... and A(r) = 3.31905014223729720342271370055697247448941708369151595... where A(r) and r satisfy A(r) = exp( (A(r)^2 + 1)/(A(r)^2 - 1) ) and r = 2*A(r)/(A(r)^2 - 1). (End)
E.g.f. A(x)=exp(B(x)), B(x) satisfies B(x)=x*cosh(B(x)). [Vladimir Kruchinin, Apr 19 2011]
a(n) ~ (1-(-1)^n*s^2)/s * n^(n-1) * ((1-s^2)/(2*s))^n / exp(n), where s = 0.3012910191606573456... is the root of the equation (1+s^2) = (s^2-1)*log(s), r = 2*s/(1-s^2). - Vaclav Kotesovec, Jan 08 2014
E.g.f. satisfies A(-x) = 1/A(x). - Alexander Burstein, Oct 26 2021
EXAMPLE
E.g.f.: A(x) = 1 + x + x^2/2! + 4x^3/3! + 13x^4/4! + 96x^5/5! +...
MAPLE
a := n -> 2^(-n)*add(binomial(n, k)*(n+1-2*k)^(n-1), k=0..n);
MATHEMATICA
a[n_] := Sum[((n-2k+1)^(n-1)*n!) / (k!*(n-k)!), {k, 0, n}] / 2^n; a[1] = 1; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Nov 14 2011, after Maple *)
PROG
(PARI) {a(n)=local(A=1+x); for(i=0, n, A=exp(x*(A+1/(A +x*O(x^n)))/2)); n!*polcoeff(A, n)} \\ Paul D. Hanna, Mar 29 2008
(PARI) {a(n) = sum(k=0, n, binomial(n, k)*(n+1-2*k)^(n-1))/2^n} \\ Seiichi Manyama, Sep 27 2020
CROSSREFS
Cf. bisections: A007106, A143601.
Cf. A138764 (variant).
KEYWORD
easy,nice,nonn
AUTHOR
Alex Postnikov (apost(AT)math.mit.edu), Nov 13 2000
EXTENSIONS
Updated URL and author's e-mail address - R. J. Mathar, May 23 2010
STATUS
approved
A370430
Expansion of e.g.f. C(x,k) satisfying C(x,k) = cosh( x*cosh(k*x*C(x,k)) ), as a triangle read by rows.
+10
7
1, 1, 0, 1, 12, 0, 1, 420, 120, 0, 1, 10248, 36400, 896, 0, 1, 196920, 4858560, 2170560, 5760, 0, 1, 3247860, 461126160, 1127738304, 102960000, 33792, 0, 1, 48361404, 35248293080, 340884800256, 187282263168, 4083183104, 186368, 0, 1, 669616080, 2290777550880, 76526954183680, 153279541958400, 25081621813248, 141360128000, 983040, 0
(list; table; graph; refs; listen; history; text; internal format)
OFFSET
0,5
COMMENTS
The row sums equal A143601, the number of labeled odd degree trees with 2n+1 nodes.
Unsigned version of triangle A370330.
A row reversal of triangle A370432.
LINKS
Paul D. Hanna, Table of n, a(n) for n = 0..1325
FORMULA
E.g.f.: C(x,k) = Sum_{n>=0} Sum_{j=0..n} a(n,j) * x^(2*n)*k^(2*j)/(2*n)! along with the related functions C = C(x,k), S = S(x,k), D = D(x,k), and T = T(x,k) satisfy the following formulas.
Definition.
(1.a) (C + S) = exp(x*D).
(1.b) (D + k*T) = exp(k*x*C).
(2.a) C^2 - S^2 = 1.
(2.b) D^2 - k^2*T^2 = 1.
Hyperbolic functions.
(3.a) C = cosh(x*D).
(3.b) S = sinh(x*D).
(3.c) D = cosh(k*x*C).
(3.d) T = (1/k) * sinh(k*x*C).
(4.a) C = cosh( x*cosh(k*x*C) ).
(4.b) S = sinh( x*cosh(k*x*sqrt(1 + S^2)) ).
(4.c) D = cosh( k*x*cosh(x*D) ).
(4.d) T = (1/k) * sinh( k*x*cosh(x*sqrt(1 + k^2*T^2)) ).
(5.a) (C*D + k*S*T) = cosh(x*D + k*x*C).
(5.b) (S*D + k*C*T) = sinh(x*D + k*x*C).
Transformations.
(6.a) C(x, 1/k) = D(x/k, k).
(6.b) D(x, 1/k) = C(x/k, k).
(6.c) S(x, 1/k) = k * T(x/k, k).
(6.d) T(x, 1/k) = k * S(x/k, k).
(6.e) D(x, k) = C(k*x, 1/k).
(6.f) C(x, k) = D(k*x, 1/k).
(6.g) T(x, k) = (1/k) * S(k*x, 1/k).
(6.h) S(x, k) = (1/k) * T(k*x, 1/k).
Integrals.
(7.a) C = 1 + Integral S*D + x*S*D' dx.
(7.b) S = Integral C*D + x*C*D' dx.
(7.c) D = 1 + k^2 * Integral T*C + x*T*C' dx.
(7.d) T = Integral D*C + x*D*C' dx.
Derivatives (d/dx).
(8.a) C*C' = S*S'.
(8.b) D*D' = k^2*T*T'.
(9.a) C' = S * (D + x*D').
(9.b) S' = C * (D + x*D').
(9.c) D' = k^2 * T * (C + x*C').
(9.d) T' = D * (C + x*C').
(10.a) C' = S * (D + k^2*x*T*C) / (1 - k^2*x^2*S*T).
(10.b) S' = C * (D + k^2*x*T*C) / (1 - k^2*x^2*S*T).
(10.c) D' = k^2 * T * (C + x*S*D) / (1 - k^2*x^2*S*T).
(10.d) T' = D * (C + x*S*D) / (1 - k^2*x^2*S*T).
(11.a) (C + x*C') = (C + x*S*D) / (1 - k^2*x^2*S*T).
(11.b) (D + x*D') = (D + k^2*x*T*C) / (1 - k^2*x^2*S*T).
Logarithms.
(12.a) D = log(C + sqrt(C^2 - 1)) / x.
(12.b) C = log(D + sqrt(D^2 - 1)) / (k*x).
(12.c) T = sqrt(log(S + sqrt(1 + S^2))^2 - x^2) / (k*x).
(12.d) S = sqrt(log(k*T + sqrt(1 + k^2*T^2))^2 - k^2*x^2) / (k*x).
EXAMPLE
E.g.f.: C(x,k) = 1 + (1)*x^2/2! + (1 + 12*k^2)*x^4/4! + (1 + 420*k^2 + 120*k^4)*x^6/6! + (1 + 10248*k^2 + 36400*k^4 + 896*k^6)*x^8/8! + (1 + 196920*k^2 + 4858560*k^4 + 2170560*k^6 + 5760*k^8)*x^10/10! + (1 + 3247860*k^2 + 461126160*k^4 + 1127738304*k^6 + 102960000*k^8 + 33792*k^10)*x^12/12! + (1 + 48361404*k^2 + 35248293080*k^4 + 340884800256*k^6 + 187282263168*k^8 + 4083183104*k^10 + 186368*k^12)*x^14/14! + ...
where C(x,k) = cosh( x*cosh(k*x*C(x,k)) ).
This triangle of coefficients a(n,j) of x^(2*n)*k^(2*j)/(2*n)! in C(x,k) begins
1;
1, 0;
1, 12, 0;
1, 420, 120, 0;
1, 10248, 36400, 896, 0;
1, 196920, 4858560, 2170560, 5760, 0;
1, 3247860, 461126160, 1127738304, 102960000, 33792, 0;
1, 48361404, 35248293080, 340884800256, 187282263168, 4083183104, 186368, 0;
1, 669616080, 2290777550880, 76526954183680, 153279541958400, 25081621813248, 141360128000, 983040, 0;
1, 8781531696, 131249560881600, 14052066349007232, 83205186217021440, 51607880705931264, 2855197025501184, 4416170065920, 5013504, 0; ...
PROG
(PARI) {a(n, j) = my(C=1, S=x, D=1, T=x, Ox=x*O(x^(2*n)));
for(i=1, 2*n,
C = cosh( x*cosh(k*x*C +Ox) );
S = sinh( x*cosh(k*x*sqrt(1 + S^2 +Ox)) );
D = cosh( k*x*cosh(x*D +Ox));
T = (1/k)*sinh( k*x*cosh(x*sqrt(1 + k^2*T^2 +Ox))); );
(2*n)! * polcoeff(polcoeff(C, 2*n, x), 2*j, k)}
for(n=0, 10, for(j=0, n, print1( a(n, j), ", ")); print(""))
CROSSREFS
Cf. A370431 (S), A370432 (D), A370433 (T), A143601 (row sums).
Cf. A370330.
KEYWORD
nonn,tabl
AUTHOR
Paul D. Hanna, Feb 19 2024
STATUS
approved
A370432
Expansion of e.g.f. D(x,k) satisfying D(x,k) = cosh( k*x*cosh(x*D(x,k)) ), as a triangle read by rows.
+10
7
1, 0, 1, 0, 12, 1, 0, 120, 420, 1, 0, 896, 36400, 10248, 1, 0, 5760, 2170560, 4858560, 196920, 1, 0, 33792, 102960000, 1127738304, 461126160, 3247860, 1, 0, 186368, 4083183104, 187282263168, 340884800256, 35248293080, 48361404, 1, 0, 983040, 141360128000, 25081621813248, 153279541958400, 76526954183680, 2290777550880, 669616080, 1
(list; table; graph; refs; listen; history; text; internal format)
OFFSET
0,5
COMMENTS
The row sums equal A143601, the number of labeled odd degree trees with 2n+1 nodes.
Unsigned version of triangle A370332.
A row reversal of triangle A370430.
LINKS
Paul D. Hanna, Table of n, a(n) for n = 0..1325
FORMULA
E.g.f.: D(x,k) = Sum_{n>=0} Sum_{j=0..n} a(n,j) * x^(2*n)*k^(2*j)/(2*n)! along with the related functions C = C(x,k), S = S(x,k), D = D(x,k), and T = T(x,k) satisfy the following formulas.
Definition.
(1.a) (C + S) = exp(x*D).
(1.b) (D + k*T) = exp(k*x*C).
(2.a) C^2 - S^2 = 1.
(2.b) D^2 - k^2*T^2 = 1.
Hyperbolic functions.
(3.a) C = cosh(x*D).
(3.b) S = sinh(x*D).
(3.c) D = cosh(k*x*C).
(3.d) T = (1/k) * sinh(k*x*C).
(4.a) C = cosh( x*cosh(k*x*C) ).
(4.b) S = sinh( x*cosh(k*x*sqrt(1 + S^2)) ).
(4.c) D = cosh( k*x*cosh(x*D) ).
(4.d) T = (1/k) * sinh( k*x*cosh(x*sqrt(1 + k^2*T^2)) ).
(5.a) (C*D + k*S*T) = cosh(x*D + k*x*C).
(5.b) (S*D + k*C*T) = sinh(x*D + k*x*C).
Transformations.
(6.a) C(x, 1/k) = D(x/k, k).
(6.b) D(x, 1/k) = C(x/k, k).
(6.c) S(x, 1/k) = k * T(x/k, k).
(6.d) T(x, 1/k) = k * S(x/k, k).
(6.e) D(x, k) = C(k*x, 1/k).
(6.f) C(x, k) = D(k*x, 1/k).
(6.g) T(x, k) = (1/k) * S(k*x, 1/k).
(6.h) S(x, k) = (1/k) * T(k*x, 1/k).
Integrals.
(7.a) C = 1 + Integral S*D + x*S*D' dx.
(7.b) S = Integral C*D + x*C*D' dx.
(7.c) D = 1 + k^2 * Integral T*C + x*T*C' dx.
(7.d) T = Integral D*C + x*D*C' dx.
Derivatives (d/dx).
(8.a) C*C' = S*S'.
(8.b) D*D' = k^2*T*T'.
(9.a) C' = S * (D + x*D').
(9.b) S' = C * (D + x*D').
(9.c) D' = k^2 * T * (C + x*C').
(9.d) T' = D * (C + x*C').
(10.a) C' = S * (D + k^2*x*T*C) / (1 - k^2*x^2*S*T).
(10.b) S' = C * (D + k^2*x*T*C) / (1 - k^2*x^2*S*T).
(10.c) D' = k^2 * T * (C + x*S*D) / (1 - k^2*x^2*S*T).
(10.d) T' = D * (C + x*S*D) / (1 - k^2*x^2*S*T).
(11.a) (C + x*C') = (C + x*S*D) / (1 - k^2*x^2*S*T).
(11.b) (D + x*D') = (D + k^2*x*T*C) / (1 - k^2*x^2*S*T).
Logarithms.
(12.a) D = log(C + sqrt(C^2 - 1)) / x.
(12.b) C = log(D + sqrt(D^2 - 1)) / (k*x).
(12.c) T = sqrt(log(S + sqrt(1 + S^2))^2 - x^2) / (k*x).
(12.d) S = sqrt(log(k*T + sqrt(1 + k^2*T^2))^2 - k^2*x^2) / (k*x).
EXAMPLE
E.g.f.: D(x,k) = 1 + (k^2)*x^2/2! + (12*k^2 + k^4)*x^4/4! + (120*k^2 + 420*k^4 + k^6)*x^6/6! + (896*k^2 + 36400*k^4 + 10248*k^6 + k^8)*x^8/8! + (5760*k^2 + 2170560*k^4 + 4858560*k^6 + 196920*k^8 + k^10)*x^10/10! + (33792*k^2 + 102960000*k^4 + 1127738304*k^6 + 461126160*k^8 + 3247860*k^10 + k^12)*x^12/12! + (186368*k^2 + 4083183104*k^4 + 187282263168*k^6 + 340884800256*k^8 + 35248293080*k^10 + 48361404*k^12 + k^14)*x^14/14! + ...
where D(x,k) = cosh( k*x*cosh(x*D(x,k)) ).
This triangle of coefficients a(n,j) of x^(2*n)*k^(2*j)/(2*n)! in D(x,k) begins
1;
0, 1;
0, 12, 1;
0, 120, 420, 1;
0, 896, 36400, 10248, 1;
0, 5760, 2170560, 4858560, 196920, 1;
0, 33792, 102960000, 1127738304, 461126160, 3247860, 1;
0, 186368, 4083183104, 187282263168, 340884800256, 35248293080, 48361404, 1;
0, 983040, 141360128000, 25081621813248, 153279541958400, 76526954183680, 2290777550880, 669616080, 1; ...
PROG
(PARI) {a(n, j) = my(C=1, S=x, D=1, T=x, Ox=x*O(x^(2*n)));
for(i=1, 2*n,
C = cosh( x*cosh(k*x*C +Ox) );
S = sinh( x*cosh(k*x*sqrt(1 + S^2 +Ox)) );
D = cosh( k*x*cosh(x*D +Ox));
T = (1/k)*sinh( k*x*cosh(x*sqrt(1 + k^2*T^2 +Ox))); );
(2*n)! * polcoeff(polcoeff(D, 2*n, x), 2*j, k)}
for(n=0, 10, for(j=0, n, print1( a(n, j), ", ")); print(""))
CROSSREFS
Cf. A370430 (C), A370431 (S), A370433 (T), A143601 (row sums).
Cf. A370332.
KEYWORD
nonn,tabl
AUTHOR
Paul D. Hanna, Feb 19 2024
STATUS
approved
A143600
E.g.f. satisfies: A(x) = exp(x*A(x)/A(-x)).
+10
5
1, 1, 5, 25, 249, 2561, 40573, 641817, 14110001, 302279617, 8530496181, 230851019609, 7964867290537, 260618470319169, 10635790073585069, 408342804482252761, 19246730825243728737, 848289638051491455617, 45356940470607637151845, 2257054105205570995111833
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 0..175
FORMULA
E.g.f. A(x) satisfies:
(1) A(x) = exp(x*exp(2x*G(2x))) where G(x) = cosh(x*G(x)) = e.g.f. of A143601.
(2) [A(x)/A(-x) + A(-x)/A(x)]/2 = G(2x) where G(x) = cosh(x*G(x)) = e.g.f. of A143601.
(3) A(x)/A(-x) = exp(x*[A(x)/A(-x) + A(-x)/A(x)]) = F(2x) where F(x) = exp(x*[F(x) + 1/F(x)]/2) = e.g.f. of A058014.
(4) A(x) = Sum_{n>=0} (n+1)^(n-1) * x^n/n! / A(-x)^n.
(5) A(x)^m = Sum_{n>=0} m*(n+m)^(n-1) * x^n/n! / A(-x)^n.
(6) log(A(x)) = Sum_{n>=1} n^(n-1) * x^n/n! / A(-x)^n = x*A(x)/A(-x).
Formulas (4), (5), and (6) are due to LambertW identities. - Paul D. Hanna, Nov 05 2012
a(n) ~ c * n! / (n^(3/2) * r^n), where r = 0.33137170967459079... is the root of the equation sqrt(1+4*r^2) = log((1+sqrt(1+4*r^2))/(2*r)), and c = 1.35397895306096963692514418... if n is even, and c = 1.281887793570420328585518150... if n is odd. - Vaclav Kotesovec, Feb 25 2014
EXAMPLE
E.g.f.: A(x) = 1 + x + 5*x^2/2! + 25*x^3/3! + 249*x^4/4! + 2561*x^5/5! +...
A LambertW identity yields the series:
A(x) = 1 + x/A(-x) + 3^1*x^2/2!/A(-x)^2 + 4^2*x^3/3!/A(-x)^3 + 5^3*x^4/4!/A(-x)^4 + 6^4*x^5/5!/A(-x)^5 +...+ (n+1)^(n-1)*x^n/n!/A(-x)^n +...
RELATED EXPANSIONS.
A(x)/A(-x) = F(2x) where F(x) is the e.g.f. of A058014:
A(x)/A(-x) = 1 + 2*x + 4*x^2/2! + 32*x^3/3! + 208*x^4/4! + 3072*x^5/5! +...
F(x) = 1 + x + 1*x^2/2! + 4*x^3/3! + 13*x^4/4! + 96*x^5/5! + 541*x^6/6! +...
which satisfies: F(x) = exp(x*(F(x) + 1/F(x))/2).
(A(x)/A(-x) + A(-x)/A(x))/2 = G(2x) where G(x) is the e.g.f. of A143601:
(A(x)/A(-x) + A(-x)/A(x))/2 = 1 + 4*x^2/2! + 208*x^4/4! + 34624*x^6/6! +...
G(x) = 1 + x^2/2! + 13*x^4/4! + 541*x^6/6! + 47545*x^8/8! +...
which satisfies G(x) = cosh(x*G(x)).
PROG
(PARI) a(n)=local(A=1+x*O(x^n)); for(i=0, n, A=exp(x*A/subst(A, x, -x))); n!*polcoeff(A, n)
(PARI) /* Formula Using a LambertW Identity: */
{a(n)=local(A=1); for(i=1, n, A=sum(k=0, n, (k+1)^(k-1)*x^k/k!/subst(A, x, -x)^k+x*O(x^n))); n!*polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", ")) \\ Paul D. Hanna, Nov 05 2012
CROSSREFS
Cf. A058014, A143601, A007106.
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Aug 26 2008
STATUS
approved
A370330
Expansion of e.g.f. C(x,k) satisfying C(x,k) = cos( x*cos(k*x*C(x,k)) ), as a triangle read by rows.
+10
4
1, -1, 0, 1, 12, 0, -1, -420, -120, 0, 1, 10248, 36400, 896, 0, -1, -196920, -4858560, -2170560, -5760, 0, 1, 3247860, 461126160, 1127738304, 102960000, 33792, 0, -1, -48361404, -35248293080, -340884800256, -187282263168, -4083183104, -186368, 0, 1, 669616080, 2290777550880, 76526954183680, 153279541958400, 25081621813248, 141360128000, 983040, 0
(list; table; graph; refs; listen; history; text; internal format)
OFFSET
0,5
COMMENTS
The unsigned row sums equal A143601.
Signed version of triangle A370430.
A row reversal of triangle A370332.
LINKS
Table of n, a(n) for n=0..44.
FORMULA
E.g.f.: C(x,k) = Sum_{n>=0} Sum_{j=0..n} a(n,j) * x^(2*n)*k^(2*j)/(2*n)! along with the related functions C = C(x,k), S = S(x,k), D = D(x,k), and T = T(x,k) satisfy the following formulas.
Definition.
(1.a) (C + i*S) = exp(i*x*D).
(1.b) (D + i*k*T) = exp(i*k*x*C).
(2.a) C^2 + S^2 = 1.
(2.b) D^2 + k^2*T^2 = 1.
Circular functions.
(3.a) C = cos(x*D).
(3.b) S = sin(x*D).
(3.c) D = cos(k*x*C).
(3.d) T = (1/k) * sin(k*x*C).
(4.a) C = cos( x*cos(k*x*C) ).
(4.b) S = sin( x*cos(k*x*sqrt(1 - S^2)) ).
(4.c) D = cos( k*x*cos(x*D) ).
(4.d) T = (1/k) * sin( k*x*cos(x*sqrt(1 - k^2*T^2)) ).
(5.a) (C*D - k*S*T) = cos(x*D + k*x*C).
(5.b) (S*D + k*C*T) = sin(x*D + k*x*C).
Transformations.
(6.a) C(x, 1/k) = D(x/k, k).
(6.b) D(x, 1/k) = C(x/k, k).
(6.c) S(x, 1/k) = k * T(x/k, k).
(6.d) T(x, 1/k) = k * S(x/k, k).
(6.e) D(x, k) = C(k*x, 1/k).
(6.f) C(x, k) = D(k*x, 1/k).
(6.g) T(x, k) = (1/k) * S(k*x, 1/k).
(6.h) S(x, k) = (1/k) * T(k*x, 1/k).
Integrals.
(7.a) C = 1 - Integral S*D + x*S*D' dx.
(7.b) S = Integral C*D + x*C*D' dx.
(7.c) D = 1 - k^2 * Integral T*C + x*T*C' dx.
(7.d) T = Integral D*C + x*D*C' dx.
Derivatives (d/dx).
(8.a) C*C' = -S*S'.
(8.b) D*D' = -k^2*T*T'.
(9.a) C' = -S * (D + x*D').
(9.b) S' = C * (D + x*D').
(9.c) D' = -k^2 * T * (C + x*C').
(9.d) T' = D * (C + x*C').
(10.a) C' = -S * (D - k^2*x*T*C) / (1 - k^2*x^2*S*T).
(10.b) S' = C * (D - k^2*x*T*C) / (1 - k^2*x^2*S*T).
(10.c) D' = -k^2 * T * (C - x*S*D) / (1 - k^2*x^2*S*T).
(10.d) T' = D * (C - x*S*D) / (1 - k^2*x^2*S*T).
(11.a) (C + x*C') = (C - x*S*D) / (1 - k^2*x^2*S*T).
(11.b) (D + x*D') = (D - k^2*x*T*C) / (1 - k^2*x^2*S*T).
EXAMPLE
E.g.f.: C(x,k) = 1 - (1)*x^2/2! + (1 + 12*k^2)*x^4/4! - (1 + 420*k^2 + 120*k^4)*x^6/6! + (1 + 10248*k^2 + 36400*k^4 + 896*k^6)*x^8/8! - (1 + 196920*k^2 + 4858560*k^4 + 2170560*k^6 + 5760*k^8)*x^10/10! + (1 + 3247860*k^2 + 461126160*k^4 + 1127738304*k^6 + 102960000*k^8 + 33792*k^10)*x^12/12! - (1 + 48361404*k^2 + 35248293080*k^4 + 340884800256*k^6 + 187282263168*k^8 + 4083183104*k^10 + 186368*k^12)*x^14/14! + ...
where C(x,k) = cos( x*cos(k*x*C(x,k)) ).
This triangle of coefficients a(n,j) of x^(2*n)*k^(2*j)/(2*n)! in C(x,k) begins
1;
-1, 0;
1, 12, 0;
-1, -420, -120, 0;
1, 10248, 36400, 896, 0;
-1, -196920, -4858560, -2170560, -5760, 0;
1, 3247860, 461126160, 1127738304, 102960000, 33792, 0;
-1, -48361404, -35248293080, -340884800256, -187282263168, -4083183104, -186368, 0;
1, 669616080, 2290777550880, 76526954183680, 153279541958400, 25081621813248, 141360128000, 983040, 0; ...
PROG
(PARI) {a(n, j) = my(C=1, S=x, D=1, T=x, Ox=x*O(x^(2*n)));
for(i=1, 2*n,
C = cos( x*cos(k*x*C +Ox) );
S = sin( x*cos(k*x*sqrt(1 - S^2 +Ox)) );
D = cos( k*x*cos(x*D +Ox));
T = (1/k)*sin( k*x*cos(x*sqrt(1 - k^2*T^2 +Ox))); );
(2*n)! * polcoeff(polcoeff(C, 2*n, x), 2*j, k)}
for(n=0, 10, for(j=0, n, print1( a(n, j), ", ")); print(""))
CROSSREFS
Cf. A370331 (S), A370332 (D), A370333 (T).
Cf. A370430.
KEYWORD
sign,tabl
AUTHOR
Paul D. Hanna, Feb 19 2024
STATUS
approved
A370332
Expansion of e.g.f. D(x,k) satisfying D(x,k) = cos( k*x*cos(x*D(x,k)) ), as a triangle read by rows.
+10
4
1, 0, -1, 0, 12, 1, 0, -120, -420, -1, 0, 896, 36400, 10248, 1, 0, -5760, -2170560, -4858560, -196920, -1, 0, 33792, 102960000, 1127738304, 461126160, 3247860, 1, 0, -186368, -4083183104, -187282263168, -340884800256, -35248293080, -48361404, -1, 0, 983040, 141360128000, 25081621813248, 153279541958400, 76526954183680, 2290777550880, 669616080, 1
(list; table; graph; refs; listen; history; text; internal format)
OFFSET
0,5
COMMENTS
The unsigned row sums equal A143601.
Signed version of triangle A370432.
A row reversal of triangle A370330.
LINKS
Table of n, a(n) for n=0..44.
FORMULA
E.g.f.: D(x,k) = Sum_{n>=0} Sum_{j=0..n} a(n,j) * x^(2*n)*k^(2*j)/(2*n)! along with the related functions C = C(x,k), S = S(x,k), D = D(x,k), and T = T(x,k) satisfy the following formulas.
Definition.
(1.a) (C + i*S) = exp(i*x*D).
(1.b) (D + i*k*T) = exp(i*k*x*C).
(2.a) C^2 + S^2 = 1.
(2.b) D^2 + k^2*T^2 = 1.
Circular functions.
(3.a) C = cos(x*D).
(3.b) S = sin(x*D).
(3.c) D = cos(k*x*C).
(3.d) T = (1/k) * sin(k*x*C).
(4.a) C = cos( x*cos(k*x*C) ).
(4.b) S = sin( x*cos(k*x*sqrt(1 - S^2)) ).
(4.c) D = cos( k*x*cos(x*D) ).
(4.d) T = (1/k) * sin( k*x*cos(x*sqrt(1 - k^2*T^2)) ).
(5.a) (C*D - k*S*T) = cos(x*D + k*x*C).
(5.b) (S*D + k*C*T) = sin(x*D + k*x*C).
Transformations.
(6.a) C(x, 1/k) = D(x/k, k).
(6.b) D(x, 1/k) = C(x/k, k).
(6.c) S(x, 1/k) = k * T(x/k, k).
(6.d) T(x, 1/k) = k * S(x/k, k).
(6.e) D(x, k) = C(k*x, 1/k).
(6.f) C(x, k) = D(k*x, 1/k).
(6.g) T(x, k) = (1/k) * S(k*x, 1/k).
(6.h) S(x, k) = (1/k) * T(k*x, 1/k).
Integrals.
(7.a) C = 1 - Integral S*D + x*S*D' dx.
(7.b) S = Integral C*D + x*C*D' dx.
(7.c) D = 1 - k^2 * Integral T*C + x*T*C' dx.
(7.d) T = Integral D*C + x*D*C' dx.
Derivatives (d/dx).
(8.a) C*C' = -S*S'.
(8.b) D*D' = -k^2*T*T'.
(9.a) C' = -S * (D + x*D').
(9.b) S' = C * (D + x*D').
(9.c) D' = -k^2 * T * (C + x*C').
(9.d) T' = D * (C + x*C').
(10.a) C' = -S * (D - k^2*x*T*C) / (1 - k^2*x^2*S*T).
(10.b) S' = C * (D - k^2*x*T*C) / (1 - k^2*x^2*S*T).
(10.c) D' = -k^2 * T * (C - x*S*D) / (1 - k^2*x^2*S*T).
(10.d) T' = D * (C - x*S*D) / (1 - k^2*x^2*S*T).
(11.a) (C + x*C') = (C - x*S*D) / (1 - k^2*x^2*S*T).
(11.b) (D + x*D') = (D - k^2*x*T*C) / (1 - k^2*x^2*S*T).
EXAMPLE
E.g.f.: D(x,k) = 1 - (k^2)*x^2/2! + (12*k^2 + k^4)*x^4/4! - (120*k^2 + 420*k^4 + k^6)*x^6/6! + (896*k^2 + 36400*k^4 + 10248*k^6 + k^8)*x^8/8! - (5760*k^2 + 2170560*k^4 + 4858560*k^6 + 196920*k^8 + k^10)*x^10/10! + (33792*k^2 + 102960000*k^4 + 1127738304*k^6 + 461126160*k^8 + 3247860*k^10 + k^12)*x^12/12! - (186368*k^2 + 4083183104*k^4 + 187282263168*k^6 + 340884800256*k^8 + 35248293080*k^10 + 48361404*k^12 + k^14)*x^14/14! + ...
where D(x,k) = cos( k*x*cos(x*D(x,k)) ).
This triangle of coefficients a(n,j) of x^(2*n)*k^(2*j)/(2*n)! in D(x,k) begins
1;
0, -1;
0, 12, 1;
0, -120, -420, -1;
0, 896, 36400, 10248, 1;
0, -5760, -2170560, -4858560, -196920, -1;
0, 33792, 102960000, 1127738304, 461126160, 3247860, 1;
0, -186368, -4083183104, -187282263168, -340884800256, -35248293080, -48361404, -1;
0, 983040, 141360128000, 25081621813248, 153279541958400, 76526954183680, 2290777550880, 669616080, 1; ...
PROG
(PARI) {a(n, j) = my(C=1, S=x, D=1, T=x, Ox=x*O(x^(2*n)));
for(i=1, 2*n,
C = cos( x*cos(k*x*C +Ox) );
S = sin( x*cos(k*x*sqrt(1 - S^2 +Ox)) );
D = cos( k*x*cos(x*D +Ox));
T = (1/k)*sin( k*x*cos(x*sqrt(1 - k^2*T^2 +Ox))); );
(2*n)! * polcoeff(polcoeff(D, 2*n, x), 2*j, k)}
for(n=0, 10, for(j=0, n, print1( a(n, j), ", ")); print(""))
CROSSREFS
Cf. A370330 (C), A370331 (S), A370333 (T).
Cf. A370432.
KEYWORD
sign,tabl
AUTHOR
Paul D. Hanna, Feb 19 2024
STATUS
approved
A372811
Expansion of e.g.f. C(x) satisfying C(x) = cosh( x*cosh(2*x*C(x)) ), where a(n) is the coefficient of x^(2*n)/(2*n)! in C(x) for n >= 0.
+10
4
1, 1, 49, 3601, 680737, 218915041, 105958624465, 74506995584113, 70436550855565633, 86815671664245679297, 135090335333407225545841, 258969022695032433287216593, 599973069857987759584855153249, 1652347283935245955005795151113121, 5336236426918250608377414155884578577
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
LINKS
Paul D. Hanna, Table of n, a(n) for n = 0..300
FORMULA
a(n) = Sum_{j=0..n} A370430(n,j) * 2^(2*j).
E.g.f.: C(x) = Sum_{n>=0} a(n) * x^(2*n)/(2*n)! along with related functions denoted by C = C(x), S = S(x), D = D(x), and T = T(x) satisfy the following formulas.
Definition.
(1.a) (C + S) = exp(x*D).
(1.b) (D + 2*T) = exp(2*x*C).
(2.a) C^2 - S^2 = 1.
(2.b) D^2 - 4*T^2 = 1.
Hyperbolic functions.
(3.a) C = cosh(x*D).
(3.b) S = sinh(x*D).
(3.c) D = cosh(2*x*C).
(3.d) T = (1/2) * sinh(2*x*C).
(4.a) C = cosh( x*cosh(2*x*C) ).
(4.b) S = sinh( x*cosh(2*x*sqrt(1 + S^2)) ).
(4.c) D = cosh( 2*x*cosh(x*D) ).
(4.d) T = (1/2) * sinh( 2*x*cosh(x*sqrt(1 + 4*T^2)) ).
(5.a) (C*D + 2*S*T) = cosh(x*D + 2*x*C).
(5.b) (S*D + 2*C*T) = sinh(x*D + 2*x*C).
Integrals.
(6.a) C = 1 + Integral S*D + x*S*D' dx.
(6.b) S = Integral C*D + x*C*D' dx.
(6.c) D = 1 + 4 * Integral T*C + x*T*C' dx.
(6.d) T = Integral D*C + x*D*C' dx.
Derivatives (d/dx).
(7.a) C*C' = S*S'.
(7.b) D*D' = 4*T*T'.
(8.a) C' = S * (D + x*D').
(8.b) S' = C * (D + x*D').
(8.c) D' = 4 * T * (C + x*C').
(8.d) T' = D * (C + x*C').
(9.a) C' = S * (D + 4*x*T*C) / (1 - 4*x^2*S*T).
(9.b) S' = C * (D + 4*x*T*C) / (1 - 4*x^2*S*T).
(9.c) D' = 4 * T * (C + x*S*D) / (1 - 4*x^2*S*T).
(9.d) T' = D * (C + x*S*D) / (1 - 4*x^2*S*T).
(10.a) (C + x*C') = (C + x*S*D) / (1 - 4*x^2*S*T).
(10.b) (D + x*D') = (D + 4*x*T*C) / (1 - 4*x^2*S*T).
Logarithms.
(11.a) D = log(C + sqrt(C^2 - 1)) / x.
(11.b) C = log(D + sqrt(D^2 - 1)) / (2*x).
(11.c) T = sqrt(log(S + sqrt(1 + S^2))^2 - x^2) / (2*x).
(11.d) S = sqrt(log(2*T + sqrt(1 + 4*T^2))^2 - 4*x^2) / (2*x).
The radius of convergence r and C(r) satisfy r = arccosh(C(r)) / cosh(2*r*C(r)) and 1 = 2*r^2 * sinh(2*r*C(r)) * sinh( r*cosh(2*r*C(r)) ), where r = 0.458693345589772637742719473602361341151810356245785213... and C(r) = 1.56301189045436141892741676499724550339640443730107995...
EXAMPLE
E.g.f: C(x) = 1 + x^2/2! + 49*x^4/4! + 3601*x^6/6! + 680737*x^8/8! + 218915041*x^10/10! + 105958624465*x^12/12! + 74506995584113*x^14/14! + ...
where C(x) = cosh( x*cosh(2*x*C(x)) ).
RELATED SERIES.
Related functions S(x), D(x), and T(x) are described below.
S(x) = x + 13*x^3/3! + 441*x^5/5! + 68069*x^7/7! + 15591025*x^9/9! + 6212017725*x^11/11! + 3652639410473*x^13/13! + 2963960104898581*x^15/15! + ...
where S(x) = sqrt(C(x)^2 - 1)
and S(x) = sinh( x*cosh( 2*x*sqrt(1 + S(x)^2) ) ).
D(x) = 1 + 4*x^2/2! + 64*x^4/4! + 7264*x^6/6! + 1242112*x^8/8! + 396112384*x^10/10! + 195196856320*x^12/12! + 135610245824512*x^14/14! + ...
where D(x) = cosh( 2*x*C(x) )
and D(x) = cosh( 2*x*cosh(x*D(x)) ).
T(x) = x + 7*x^3/3! + 381*x^5/5! + 50051*x^7/7! + 11899705*x^9/9! + 4787171775*x^11/11! + 2800735142453*x^13/13! + 2286983798222779*x^15/15! + ...
where T(x) = (1/2) * sinh( 2*x*C(x) )
and T(x) = (1/2) * sinh( 2*x*cosh( x*sqrt(1 + 4*T(x)^2) ) ).
SPECIFIC VALUES.
C(1/3) = 1.09195477630888952286167165173829275218422327069159600...
C(1/4) = 1.04077887287196699205886551058236704630676681245696946...
C(1/5) = 1.02363722685574465853941118194990482596731360834136139...
C(1/6) = 1.01558260957952973250484327981285267794556192126351939...
C(1/10) = 1.00520934315195311495083109289140774006817550223233669...
PROG
(PARI) /* From C(x) = cosh( x*cosh(2*x*C(x)) ) */
{a(n) = my(C=1); for(i=0, n, C=truncate(C); C = cosh( x*cosh(2*x*C + x*O(x^(2*i))) ));
(2*n)! * polcoeff(C, 2*n, x)}
for(n=0, 30, print1( a(n), ", "))
(PARI) /* From A370430 at k = 2 */
{a(n, k = 2) = my(C=1, S=x, D=1, T=x, Ox=x*O(x^(2*n)));
for(i=1, 2*n,
C = cosh( x*cosh(k*x*C +Ox) );
S = sinh( x*cosh(k*x*sqrt(1 + S^2 +Ox)) );
D = cosh( k*x*cosh(x*D +Ox));
T = (1/k)*sinh( k*x*cosh(x*sqrt(1 + k^2*T^2 +Ox))); );
(2*n)! * polcoeff(C, 2*n, x)}
for(n=0, 30, print1( a(n), ", "))
CROSSREFS
Cf. A370430 (k = 2), A372812 (S(x)), A372813 (D(x)), A372814 (T(x)), A143601.
KEYWORD
nonn
AUTHOR
Paul D. Hanna, May 16 2024
STATUS
approved
A372813
Expansion of e.g.f. D(x) satisfying D(x) = cosh( 2*x*cosh(x*D(x)) ), where a(n) is the coefficient of x^(2*n)/(2*n)! in D(x) for n >= 0.
+10
4
1, 4, 64, 7264, 1242112, 396112384, 195196856320, 135610245824512, 128604645225791488, 158304763492800790528, 246175295718345884041216, 471837283882871579572436992, 1092672848842771034323176914944, 3008542003438261199300841957228544, 9713742135846618809223753670120701952
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
LINKS
Paul D. Hanna, Table of n, a(n) for n = 0..300
FORMULA
a(n) = Sum_{j=0..n} A370432(n,j) * 2^(2*j).
E.g.f.: D(x) = Sum_{n>=0} a(n) * x^(2*n)/(2*n)! along with related functions denoted by C = C(x), S = S(x), D = D(x), and T = T(x) satisfy the following formulas.
Definition.
(1.a) (C + S) = exp(x*D).
(1.b) (D + 2*T) = exp(2*x*C).
(2.a) C^2 - S^2 = 1.
(2.b) D^2 - 4*T^2 = 1.
Hyperbolic functions.
(3.a) C = cosh(x*D).
(3.b) S = sinh(x*D).
(3.c) D = cosh(2*x*C).
(3.d) T = (1/2) * sinh(2*x*C).
(4.a) C = cosh( x*cosh(2*x*C) ).
(4.b) S = sinh( x*cosh(2*x*sqrt(1 + S^2)) ).
(4.c) D = cosh( 2*x*cosh(x*D) ).
(4.d) T = (1/2) * sinh( 2*x*cosh(x*sqrt(1 + 4*T^2)) ).
(5.a) (C*D + 2*S*T) = cosh(x*D + 2*x*C).
(5.b) (S*D + 2*C*T) = sinh(x*D + 2*x*C).
Integrals.
(6.a) C = 1 + Integral S*D + x*S*D' dx.
(6.b) S = Integral C*D + x*C*D' dx.
(6.c) D = 1 + 4 * Integral T*C + x*T*C' dx.
(6.d) T = Integral D*C + x*D*C' dx.
Derivatives (d/dx).
(7.a) C*C' = S*S'.
(7.b) D*D' = 4*T*T'.
(8.a) C' = S * (D + x*D').
(8.b) S' = C * (D + x*D').
(8.c) D' = 4 * T * (C + x*C').
(8.d) T' = D * (C + x*C').
(9.a) C' = S * (D + 4*x*T*C) / (1 - 4*x^2*S*T).
(9.b) S' = C * (D + 4*x*T*C) / (1 - 4*x^2*S*T).
(9.c) D' = 4 * T * (C + x*S*D) / (1 - 4*x^2*S*T).
(9.d) T' = D * (C + x*S*D) / (1 - 4*x^2*S*T).
(10.a) (C + x*C') = (C + x*S*D) / (1 - 4*x^2*S*T).
(10.b) (D + x*D') = (D + 4*x*T*C) / (1 - 4*x^2*S*T).
Logarithms.
(11.a) D = log(C + sqrt(C^2 - 1)) / x.
(11.b) C = log(D + sqrt(D^2 - 1)) / (2*x).
(11.c) T = sqrt(log(S + sqrt(1 + S^2))^2 - x^2) / (2*x).
(11.d) S = sqrt(log(2*T + sqrt(1 + 4*T^2))^2 - 4*x^2) / (2*x).
The radius of convergence r of e.g.f. D(x) is r = 0.458693345589772637742719473602361341151810356245785213... where D(r) = 2.216675597008249888019540624981069492182564304724769248...
EXAMPLE
E.g.f: D(x) = 1 + 4*x^2/2! + 64*x^4/4! + 7264*x^6/6! + 1242112*x^8/8! + 396112384*x^10/10! + 195196856320*x^12/12! + 135610245824512*x^14/14! + ...
and D(x) = cosh( 2*x*cosh(x*D(x)) ).
RELATED SERIES.
Related functions C(x), S(x), and T(x) are described below.
C(x) = 1 + x^2/2! + 49*x^4/4! + 3601*x^6/6! + 680737*x^8/8! + 218915041*x^10/10! + 105958624465*x^12/12! + 74506995584113*x^14/14! + ...
where C = cosh(x*D)
and C(x) = cosh( x*cosh(2*x*C(x)) ).
S(x) = x + 13*x^3/3! + 441*x^5/5! + 68069*x^7/7! + 15591025*x^9/9! + 6212017725*x^11/11! + 3652639410473*x^13/13! + 2963960104898581*x^15/15! + ...
where S(x) = S = sinh(x*D)
and S(x) = sinh( x*cosh( 2*x*sqrt(1 + S(x)^2) ) ).
T(x) = x + 7*x^3/3! + 381*x^5/5! + 50051*x^7/7! + 11899705*x^9/9! + 4787171775*x^11/11! + 2800735142453*x^13/13! + 2286983798222779*x^15/15! + ...
where T(x) = (1/2) * sqrt(D^2 - 1)
and T(x) = (1/2) * sinh( 2*x*cosh( x*sqrt(1 + 4*T(x)^2) ) ).
SPECIFIC VALUES.
D(1/3) = 1.276880244449228122993163054974488376796865611992370031...
D(1/4) = 1.138485942600540714616500323386982626365733417421170976...
D(1/5) = 1.085004369634098854421041251800873218914671999144038407...
D(1/6) = 1.057849764714936388260012199112395774792001649565003101...
D(1/10) = 1.020277074958546717842943931766605150247847706664020751...
PROG
(PARI) /* From D(x) = cosh( 2*x*cosh(x*D(x)) ) */
{a(n) = my(D=1); for(i=0, n, D=truncate(D); D = cosh( 2*x*cosh(x*D + x*O(x^(2*i))) ));
(2*n)! * polcoeff(D, 2*n, x)}
for(n=0, 30, print1( a(n), ", "))
(PARI) /* From A370432 at k = 2 */
{a(n, k = 2) = my(C=1, S=x, D=1, T=x, Ox=x*O(x^(2*n)));
for(i=1, 2*n,
C = cosh( x*cosh(k*x*C +Ox) );
S = sinh( x*cosh(k*x*sqrt(1 + S^2 +Ox)) );
D = cosh( k*x*cosh(x*D +Ox));
T = (1/k)*sinh( k*x*cosh(x*sqrt(1 + k^2*T^2 +Ox))); );
(2*n)! * polcoeff(D, 2*n, x)}
for(n=0, 30, print1( a(n), ", "))
CROSSREFS
Cf. A370432 (k = 2), A372811 (C(x)), A372812 (S(x)), A372814 (T(x)), A143601.
KEYWORD
nonn
AUTHOR
Paul D. Hanna, May 16 2024
STATUS
approved
page 1 2

Search completed in 0.016 seconds

Lookup Welcome Wiki Register Music Plot 2 Demos Index WebCam Contribute Format Style Sheet Transforms Superseeker Recents
The OEIS Community
Maintained by The OEIS Foundation Inc.
Last modified October 5 00:39 EDT 2024. Contains 376527 sequences.
License Agreements, Terms of Use, Privacy Policy