A306042 Expansion of e.g.f. Product_{k>=1} 1/(1 - log(1 + x)^k).

1, 1, 3, 8, 50, 94, 2446, -9024, 297216, -3183264, 64191984, -1041792192, 22098943632, -478805234064, 11856288460272, -308662348027008, 8575865689645440, -248582819381690880, 7556655091130023680, -240521346554744194560, 8049494171497089265920, -283469026458500121634560

Offset: 0

Author

Ilya Gutkovskiy, Jun 17 2018

Keywords

• sign

Links

• N. J. A. Sloane, Transforms
• Eric Weisstein's World of Mathematics, Stirling Transform

Crossrefs

Cf. A000041, A006252, A053529, A167137, A320349.

Programs

Maple

a:=series(mul(1/(1-log(1+x)^k),k=1..100),x=0,22): seq(n!*coeff(a,x,n),n=0..21); # Paolo P. Lava, Mar 26 2019

Mathematica

nmax = 21; CoefficientList[Series[Product[1/(1 - Log[1 + x]^k), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 21; CoefficientList[Series[Exp[Sum[DivisorSigma[1, k] Log[1 + x]^k/k, {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[StirlingS1[n, k] PartitionsP[k] k!, {k, 0, n}], {n, 0, 21}]

Formula

E.g.f.: exp(Sum_{k>=1} sigma(k)*log(1 + x)^k/k).

a(n) = Sum_{k=0..n} Stirling1(n,k)*A000041(k)*k!.