A330047 Expansion of e.g.f. exp(-x) / (1 - sinh(x)).

1, 0, 1, 3, 13, 75, 511, 4053, 36793, 375735, 4262971, 53203953, 724379173, 10684377795, 169713810631, 2888340723453, 52433443111153, 1011340189494255, 20654264750645491, 445249365444296553, 10103533212012216733, 240731286454287293115, 6008902898851584479551

Offset: 0

Author

Ilya Gutkovskiy, Nov 28 2019

Keywords

• nonn

Comments

Inverse binomial transform of A006154.

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..440

Crossrefs

Cf. A006154, A009227, A052841, A330046, A346894, A346895, A346921.

Programs

Mathematica

nmax = 22; CoefficientList[Series[Exp[-x]/(1 - Sinh[x]), {x, 0, nmax}], x] Range[0, nmax]!

PARI

my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1-(exp(x)-1)^2/2))) \\ Seiichi Manyama, May 07 2022

PARI

my(N=30, x='x+O('x^N)); Vec(sum(k=0, N, (2*k)!*x^(2*k)/(2^k*prod(j=1, 2*k, 1-j*x)))) \\ Seiichi Manyama, May 07 2022

PARI

a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=1, i, binomial(i, j)*stirling(j, 2, 2)*v[i-j+1])); v; \\ Seiichi Manyama, May 07 2022

PARI

a(n) = sum(k=0, n\2, (2*k)!*stirling(n, 2*k, 2)/2^k); \\ Seiichi Manyama, May 07 2022

Formula

a(n) = Sum_{k=0..n} (-1)^(n - k) * binomial(n,k) * A006154(k).

a(n) ~ n! / ((2 + sqrt(2)) * (log(1 + sqrt(2)))^(n+1)). - Vaclav Kotesovec, Dec 03 2019

From Seiichi Manyama, May 07 2022: (Start)

E.g.f.: 1/(1 - (exp(x) - 1)^2 / 2).

G.f.: Sum_{k>=0} (2*k)! * x^(2*k)/(2^k * Product_{j=1..2*k} (1 - j * x)).

a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k) * Stirling2(k,2) * a(n-k).

a(n) = Sum_{k=0..floor(n/2)} (2*k)! * Stirling2(n,2*k)/2^k. (End)

a(0) = 1; a(n) = (-1)^n + Sum_{k=1..ceiling(n/2)} binomial(n,2*k-1) * a(n-2*k+1). - Prabha Sivaramannair, Oct 06 2023