A351736 Expansion of e.g.f. exp( x * (exp(2 * x) - 1) ).

1, 0, 4, 12, 80, 560, 4512, 40768, 407808, 4453632, 52605440, 667234304, 9032423424, 129822564352, 1972450443264, 31559866736640, 530043925495808, 9317136303718400, 170976603113127936, 3268020569256755200, 64928967058257346560, 1338431135849666052096

Offset

0,3

Links

Seiichi Manyama, Table of n, a(n) for n = 0..488

Formula

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

From Seiichi Manyama, Aug 29 2022: (Start)

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

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

Prog

(PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(x*(exp(2*x)-1))))
(PARI) a(n) = n!*sum(k=0, n\2, 2^(n-k)*stirling(n-k, k, 2)/(n-k)!);
(PARI) a(n) = sum(k=0, n, (2*k-1)^(n-k)*binomial(n, k)); \\ Seiichi Manyama, Aug 29 2022
(PARI) my(N=30, x='x+O('x^N)); Vec(sum(k=0, N, x^k/(1-(2*k-1)*x)^(k+1))) \\ Seiichi Manyama, Aug 29 2022

Crossrefs

Cf. A052506, A351737.

Cf. A053491, A351733.

Sequence in context: A165261 A027145 A299795 * A010370 A197852 A362384

Adjacent sequences: A351733 A351734 A351735 * A351737 A351738 A351739

Keyword

nonn

Author

Seiichi Manyama, May 20 2022

Status

approved