A235134 Expansion of e.g.f. 1/(1 - sinh(2*x))^(1/2).

1, 1, 3, 19, 153, 1561, 19563, 289339, 4932273, 95258161, 2055639123, 49019157859, 1280056939593, 36329281202761, 1113449691889083, 36651273215389579, 1289577677407798113, 48299079453732363361, 1918528841276621473443, 80559757274836073592499

Offset: 0

Author

Vaclav Kotesovec, Jan 03 2014

Keywords

• nonn
• easy

Comments

Generally, for e.g.f. 1/(1-sinh(p*x))^(1/p) we have a(n) ~ n! * p^n / (Gamma(1/p) * 2^(1/(2*p)) * n^(1-1/p) * (arcsinh(1))^(n+1/p)).

Links

• G. C. Greubel, Table of n, a(n) for n = 0..395

Crossrefs

Cf. A006154, A235135.

Cf. A001147, A001586, A136630, A235131, A380155.

Programs

Mathematica

CoefficientList[Series[1/(1-Sinh[2*x])^(1/2), {x, 0, 20}], x] * Range[0, 20]!

PARI

x='x+O('x^50); Vec(serlaplace(1/(sqrt(1-sinh(2*x))))) \\ G. C. Greubel, Apr 05 2017

PARI

a136630(n, k) = 1/(2^k*k!)*sum(j=0, k, (-1)^(k-j)*(2*j-k)^n*binomial(k, j));
a001147(n) = prod(k=0, n-1, 2*k+1);
a(n) = sum(k=0, n, a001147(k)*2^(n-k)*a136630(n, k)); \\ Seiichi Manyama, Jun 24 2025

Formula

a(n) ~ n! * 2^(n-1/4) / (sqrt(Pi*n) * (log(1+sqrt(2)))^(n+1/2)).

a(n) = Sum_{k=0..n} A001147(k) * 2^(n-k) * A136630(n,k). - Seiichi Manyama, Jun 24 2025