A318002 E.g.f.: 2*cosh(x) / (1 + sqrt(1 - 2*sinh(2*x))).

1, 1, 5, 37, 425, 6601, 129005, 3044077, 84239825, 2675886481, 95979282005, 3837251617717, 169216980911225, 8160026826620761, 427179965967027005, 24127907244206776957, 1462542541799076574625, 94704025153744512625441, 6524332029969395884644005, 476487260493293293849001797, 36772596077297424381362590025, 2990260766874609440239439756521

Offset

0,3

Links

Table of n, a(n) for n=0..21.

Formula

E.g.f. A(x) satisfies:

(1a) A(x) = cosh(x) + sinh(x)*A(x)^2.

(1b) A(x) = cosh(x) * Sum_{n>=0} binomial(2*n,n)/(n+1) * sinh(2*x)^n/2^n.

(1c) A(x) = (1 - sqrt(1 - 2*sinh(2*x))) / (2*sinh(x)).

(2) A( -log(A(x)) ) = exp(-x).

(3a) 1 = cosh(x + log(A(x))) + sinh(x - log(A(x))).

(3b) 1 = Sum_{n>=0} ( x + (-1)^n*log(A(x)) )^n/n!.

a(n) ~ 5^(1/4) * phi^(3/2) * 2^(n - 1/2) * n^(n-1) / (exp(n) * log(phi)^(n - 1/2)), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Aug 21 2018

Example

E.g.f.: A(x) = 1 + x + 5*x^2/2! + 37*x^3/3! + 425*x^4/4! + 6601*x^5/5! + 129005*x^6/6! + 3044077*x^7/7! + 84239825*x^8/8! + 2675886481*x^9/9! + ...
such that
A(x) = cosh(x) + sinh(x)*A(x)^2.
RELATED SERIES.
log(A(x)) = x + 4*x^2/2! + 24*x^3/3! + 256*x^4/4! + 3840*x^5/5! + 73024*x^6/6! + 1688064*x^7/7! + 45991936*x^8/8! + ... + A318000(n)*x^n/n! + ...
where A( -log(A(x)) ) = exp(-x).
A(x)^2 = 1 + 2*x + 12*x^2/2! + 104*x^3/3! + 1296*x^4/4! + 21152*x^5/5! + 428352*x^6/6! + 10381184*x^7/7! + 293304576*x^8/8! + 9472819712*x^9/9! + ...

Mathematica

With[{nn=30}, CoefficientList[Series[(2 Cosh[x])/(1+Sqrt[1-2Sinh[2x]]), {x, 0, nn}], x] Range[0, nn]!] (* Harvey P. Dale, Oct 31 2024 *)

Prog

(PARI) {a(n) = my(A = 2*cosh(x +x^2*O(x^n)) / (1 + sqrt(1 - 2*sinh(2*x +x^2*O(x^n)))) ); n!*polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", "))

Crossrefs

Cf. A318000 (log(A(x))).

Sequence in context: A292873 A161565 A235345 * A323567 A304865 A003709

Adjacent sequences: A317999 A318000 A318001 * A318003 A318004 A318005

Keyword

nonn

Author

Paul D. Hanna, Aug 20 2018

Status

approved