A330047 - OEIS

%I #28 Oct 07 2023 08:48:02

%S 1,0,1,3,13,75,511,4053,36793,375735,4262971,53203953,724379173,

%T 10684377795,169713810631,2888340723453,52433443111153,

%U 1011340189494255,20654264750645491,445249365444296553,10103533212012216733,240731286454287293115,6008902898851584479551

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

%C Inverse binomial transform of A006154.

%H Seiichi Manyama, <a href="/A330047/b330047.txt">Table of n, a(n) for n = 0..440</a>

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

%F a(n) ~ n! / ((2 + sqrt(2)) * (log(1 + sqrt(2)))^(n+1)). - _Vaclav Kotesovec_, Dec 03 2019

%F From _Seiichi Manyama_, May 07 2022: (Start)

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

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

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

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

%F 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

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

%o (PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1-(exp(x)-1)^2/2))) \\ _Seiichi Manyama_, May 07 2022

%o (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

%o (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

%o (PARI) a(n) = sum(k=0, n\2, (2*k)!*stirling(n, 2*k, 2)/2^k); \\ _Seiichi Manyama_, May 07 2022

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

%K nonn

%O 0,4

%A _Ilya Gutkovskiy_, Nov 28 2019