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

%I A298905 #16 Feb 16 2025 08:33:53
%S A298905 1,1,1,8,-8,224,-712,9120,-53496,980088,-14394648,264140832,
%T A298905 -4113747024,59028225840,-545558201424,-4191307074432,450100910950272,
%U A298905 -17302659472138752,530508727766191104,-14790496500550616832,408513443917280375808,-12274212131738107257600
%N A298905 Expansion of e.g.f. Product_{k>=1} (1 + log(1 + x)^k).
%H A298905 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%H A298905 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/StirlingTransform.html">Stirling Transform</a>
%F A298905 E.g.f.: exp(Sum_{k>=1} (-1)^(k+1)*log(1 + x)^k/(k*(1 - log(1 + x)^k))).
%F A298905 a(n) = Sum_{k=0..n} Stirling1(n,k)*A000009(k)*k!.
%p A298905 b:= proc(n) option remember; `if`(n=0, 1, add(b(n-j)*add(
%p A298905      `if`(d::odd, d, 0), d=numtheory[divisors](j)), j=1..n)/n)
%p A298905     end:
%p A298905 a:= n-> add(Stirling1(n, j)*b(j)*j!, j=0..n):
%p A298905 seq(a(n), n=0..23);  # _Alois P. Heinz_, Jun 18 2018
%t A298905 nmax = 21; CoefficientList[Series[Product[(1 + Log[1 + x]^k), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
%t A298905 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]!
%t A298905 Table[Sum[StirlingS1[n, k] PartitionsQ[k] k!, {k, 0, n}], {n, 0, 21}]
%Y A298905 Cf. A000009, A088311, A305550, A306023, A306042, A320350.
%K A298905 sign
%O A298905 0,4
%A A298905 _Ilya Gutkovskiy_, Jun 18 2018