A235135 Expansion of e.g.f. 1/(1 - sinh(3*x))^(1/3).

1, 1, 4, 37, 424, 6241, 113824, 2460277, 61504384, 1746727201, 55545439744, 1955176596517, 75470959673344, 3169939381277761, 143927870364811264, 7024566555751464757, 366742587098140770304, 20394984041632355113921, 1203587891190987380752384, 75124090160952970927512997

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)).

Crossrefs

Cf. A006154, A235134.

Cf. A007559, A007788, A136630, A235132.

Programs

Mathematica

CoefficientList[Series[1/(1-Sinh[3*x])^(1/3), {x, 0, 20}], x] * Range[0, 20]!

PARI

a136630(n, k) = 1/(2^k*k!)*sum(j=0, k, (-1)^(k-j)*(2*j-k)^n*binomial(k, j));
a007559(n) = prod(k=0, n-1, 3*k+1);
a(n) = sum(k=0, n, a007559(k)*3^(n-k)*a136630(n, k)); \\ Seiichi Manyama, Jun 24 2025

Formula

a(n) ~ n! * 3^n / (Gamma(1/3) * 2^(1/6) * n^(2/3) * (log(1+sqrt(2)))^(n+1/3)).

a(n) = Sum_{k=0..n} A007559(k) * 3^(n-k) * A136630(n,k). - Seiichi Manyama, Jun 24 2025