A354325 Expansion of e.g.f. 1/(1 - x/4 * (exp(2 * x) - 1)).

1, 0, 1, 3, 14, 80, 558, 4522, 41864, 436032, 5046680, 64251176, 892361520, 13426491520, 217555171568, 3776935252560, 69942048682112, 1376150998836224, 28669321699355520, 630448829825395840, 14593473117397510400, 354696400190943197184, 9031466708133617225984

Offset

0,4

Links

Table of n, a(n) for n=0..22.

Formula

a(0) = 1; a(n) = Sum_{k=2..n} k * 2^(k-3) * binomial(n,k) * a(n-k).

a(n) = n! * Sum_{k=0..floor(n/2)} 2^(n-3*k) * k! * Stirling2(n-k,k)/(n-k)!.

Mathematica

With[{nn=30}, CoefficientList[Series[1/(1-x/4 (Exp[2x]-1)), {x, 0, nn}], x] Range[0, nn]!] (* Harvey P. Dale, Dec 02 2022 *)

Prog

(PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1-x/4*(exp(2*x)-1))))
(PARI) a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=2, i, j*2^(j-3)*binomial(i, j)*v[i-j+1])); v;
(PARI) a(n) = n!*sum(k=0, n\2, 2^(n-3*k)*k!*stirling(n-k, k, 2)/(n-k)!);

Crossrefs

Cf. A354313, A354323, A354326.

Sequence in context: A009053 A202474 A347001 * A346966 A256336 A256338

Adjacent sequences: A354322 A354323 A354324 * A354326 A354327 A354328

Keyword

nonn

Author

Seiichi Manyama, May 24 2022

Status

approved