A354121 Expansion of e.g.f. 1/(1 - log(1 + x))^4.

1, 4, 16, 68, 316, 1616, 9080, 55800, 373080, 2699520, 21035040, 175708320, 1566916320, 14862171840, 149429426880, 1587766126080, 17779538050560, 209295747832320, 2583920845209600, 33389139008678400, 450642388471395840, 6342869733912760320

Offset: 0

Author

Seiichi Manyama, May 17 2022

Keywords

• sign

Comments

a(46) is negative. - Vaclav Kotesovec, Jun 04 2022

It appears that a(n) is negative for even n >= 46. - Felix Fröhlich, Jun 04 2022

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..450

Crossrefs

Cf. A006252, A317280, A354120.

Cf. A226738, A354123.

Programs

Mathematica

Table[Sum[(k+3)! * StirlingS1[n,k], {k,0,n}]/6, {n,0,20}] (* Vaclav Kotesovec, Jun 04 2022 *)

PARI

my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1-log(1+x))^4))

PARI

a(n) = sum(k=0, n, (k+3)!*stirling(n, k, 1))/6;

Formula

a(n) = (1/6) * Sum_{k=0..n} (k + 3)! * Stirling1(n,k).

a(0) = 1; a(n) = Sum_{k=1..n} (-1)^(k-1) * (3 * k/n + 1) * (k-1)! * binomial(n,k) * a(n-k). - Seiichi Manyama, Nov 19 2023