A357598 Expansion of e.g.f. sinh(2 * (exp(x)-1)) / 2.

0, 1, 1, 5, 25, 117, 601, 3509, 22457, 153141, 1105561, 8453557, 68339833, 581495605, 5184047961, 48259748533, 468040609593, 4719817792565, 49396003390489, 535526127566773, 6004124908829177, 69509047405180213, 829801009239621849, 10202835010223731893

Offset: 0

Author

Seiichi Manyama, Oct 05 2022

Keywords

• nonn

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..558
• Eric Weisstein's World of Mathematics, Bell Polynomial.

Crossrefs

Cf. A024429, A264037, A357572.

Cf. A065143, A078944, A357599.

Programs

PARI

my(N=30, x='x+O('x^N)); concat(0, Vec(serlaplace(sinh(2*(exp(x)-1))/2)))

PARI

a(n) = sum(k=0, (n-1)\2, 4^k*stirling(n, 2*k+1, 2));

PARI

Bell_poly(n, x) = exp(-x)*suminf(k=0, k^n*x^k/k!);
a(n) = round((Bell_poly(n, 2)-Bell_poly(n, -2)))/4;

Formula

a(n) = Sum_{k=0..floor((n-1)/2)} 4^k * Stirling2(n,2*k+1).

a(n) = ( Bell_n(2) - Bell_n(-2) )/4, where Bell_n(x) is n-th Bell polynomial.

a(n) = 0; a(n) = Sum_{k=0..n-1} binomial(n-1, k) * A065143(k).