A354120 Expansion of e.g.f. 1/(1 - log(1 + x))^3.

1, 3, 9, 30, 114, 492, 2388, 12912, 77016, 503112, 3570552, 27399600, 225729360, 1991996640, 18690559200, 186620451840, 1963991600640, 21914748541440, 255336518292480, 3155705206364160, 40209018105116160, 547746803311864320, 7525926332189130240

Offset: 0

Author

Seiichi Manyama, May 17 2022

Keywords

• sign

Comments

a(34) is negative. - Vaclav Kotesovec, Jun 04 2022

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..449

Crossrefs

Cf. A006252, A317280, A354121.

Cf. A226515, A354122.

Programs

Mathematica

Table[Sum[(k+2)! * StirlingS1[n,k], {k,0,n}]/2, {n,0,35}] (* Vaclav Kotesovec, Jun 04 2022 *)
With[{nn=30},CoefficientList[Series[1/(1-Log[1+x])^3,{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, May 16 2025 *)

PARI

my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1-log(1+x))^3))

PARI

a(n) = sum(k=0, n, (k+2)!*stirling(n, k, 1))/2;

Formula

a(n) = (1/2) * Sum_{k=0..n} (k + 2)! * Stirling1(n,k).

a(0) = 1; a(n) = Sum_{k=1..n} (-1)^(k-1) * (2 * k/n + 1) * (k-1)! * binomial(n,k) * a(n-k). - Seiichi Manyama, Nov 19 2023