A354250 Expansion of e.g.f. Sum_{k>=0} (3*k)! * log(1+x)^k / k!.

1, 6, 714, 360732, 476832204, 1302897016944, 6382799223892560, 50956720815425427360, 619019914356960664044960, 10866561174598537960652828160, 264763399994627082733034386813440, 8668743073576807048450006051943930880

Offset

0,2

Comments

In general, for m > 1, Sum_{k=0..n} (m*k)! * Stirling1(n,k) ~ c * (m*n)!, where c = exp(-1/8) if m = 2, and c = 1 if m > 2. - Vaclav Kotesovec, Jan 24 2026

Links

Table of n, a(n) for n=0..11.

Formula

a(n) = Sum_{k=0..n} (3*k)! * Stirling1(n,k).

a(n) ~ sqrt(2*Pi) * 3^(3*n + 1/2) * n^(3*n + 1/2) / exp(3*n). - Vaclav Kotesovec, Jan 24 2026

Mathematica

Table[Sum[(3*k)!*StirlingS1[n, k], {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Jan 24 2026 *)

Prog

(PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (3*k)!*log(1+x)^k/k!)))
(PARI) a(n) = sum(k=0, n, (3*k)!*stirling(n, k, 1));

Crossrefs

Cf. A006252, A354243.

Cf. A316748, A354229, A354251.

Sequence in context: A112637 A289366 A253120 * A218559 A201391 A374331

Adjacent sequences: A354247 A354248 A354249 * A354251 A354252 A354253

Keyword

nonn

Author

Seiichi Manyama, May 21 2022

Status

approved