A298905 Expansion of e.g.f. Product_{k>=1} (1 + log(1 + x)^k).

1, 1, 1, 8, -8, 224, -712, 9120, -53496, 980088, -14394648, 264140832, -4113747024, 59028225840, -545558201424, -4191307074432, 450100910950272, -17302659472138752, 530508727766191104, -14790496500550616832, 408513443917280375808, -12274212131738107257600

Offset: 0

Author

Ilya Gutkovskiy, Jun 18 2018

Keywords

• sign

Links

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

Crossrefs

Cf. A000009, A088311, A305550, A306023, A306042, A320350.

Programs

Maple

b:= proc(n) option remember; `if`(n=0, 1, add(b(n-j)*add(
     `if`(d::odd, d, 0), d=numtheory[divisors](j)), j=1..n)/n)
    end:
a:= n-> add(Stirling1(n, j)*b(j)*j!, j=0..n):
seq(a(n), n=0..23);  # Alois P. Heinz, Jun 18 2018

Mathematica

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

Formula

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

a(n) = Sum_{k=0..n} Stirling1(n,k)*A000009(k)*k!.