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URL: https://oeis.org/A298905

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A298905
Expansion of e.g.f. Product_{k>=1} (1 + log(1 + x)^k).
3
1, 1, 1, 8, -8, 224, -712, 9120, -53496, 980088, -14394648, 264140832, -4113747024, 59028225840, -545558201424, -4191307074432, 450100910950272, -17302659472138752, 530508727766191104, -14790496500550616832, 408513443917280375808, -12274212131738107257600
OFFSET
0,4
LINKS
N. J. A. Sloane, Transforms
Eric Weisstein's World of Mathematics, Stirling Transform
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!.
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}]
CROSSREFS
KEYWORD
sign
AUTHOR
Ilya Gutkovskiy, Jun 18 2018
STATUS
approved