A354323 Expansion of e.g.f. exp( x/4 * (exp(2 * x) - 1) ).

1, 0, 1, 3, 11, 50, 273, 1687, 11505, 86004, 700445, 6163751, 58148547, 584622766, 6235669629, 70286727435, 834288853217, 10395375065096, 135592878107673, 1846897191981835, 26212412703559515, 386874121137659274, 5927186655133112105, 94108950154465139807

Offset

0,4

Links

Table of n, a(n) for n=0..23.

Formula

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

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

Prog

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

Crossrefs

Cf. A354325.

Sequence in context: A323672 A103466 A346762 * A230961 A203166 A000254

Adjacent sequences: A354320 A354321 A354322 * A354324 A354325 A354326

Keyword

nonn

Author

Seiichi Manyama, May 24 2022

Status

approved