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%I #9 Dec 15 2019 02:41:14
%S 1,2,12,62,420,3782,40572,463262,5708820,80773622,1319927532,
%T 23675250062,447145154820,8830952572262,185694817024092,
%U 4246473212654462,105754322266866420,2811068529133151702,78039884046777282252,2243558766132057764462
%N Expansion of e.g.f. Sum_{k>=1} (-1)^(k + 1) * (exp(x) - 1)^k / (k * (1 - (exp(x) - 1)^k)).
%F E.g.f.: -Sum_{k>=1} log(1 - (exp(x) - 1)^(2*k - 1)).
%F E.g.f.: A(x) = log(B(x)), where B(x) = e.g.f. of A305550.
%F exp(Sum_{n>=1} a(n) * log(1 + x)^n / n!) = g.f. of A000009.
%F a(n) = Sum_{k=1..n} Stirling2(n,k) * (k - 1)! * A000593(k).
%F E.g.f.: Sum_{k>=1} log(1 + (exp(x) - 1)^k). - _Vaclav Kotesovec_, Dec 15 2019
%F a(n) ~ n! * Pi^2 / (24 * (log(2))^(n+1)). - _Vaclav Kotesovec_, Dec 15 2019
%t nmax = 20; CoefficientList[Series[Sum[(-1)^(k + 1) (Exp[x] - 1)^k/(k (1 - (Exp[x] - 1)^k)), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]! // Rest
%t Table[Sum[StirlingS2[n, k] (k - 1)! Sum[Mod[d, 2] d, {d, Divisors[k]}], {k, 1, n}], {n, 1, 20}]
%t nmax = 20; Rest[CoefficientList[Series[Sum[Log[1 + (Exp[x] - 1)^k], {k, 1, nmax}], {x, 0, nmax}], x] * Range[0, nmax]!] (* _Vaclav Kotesovec_, Dec 15 2019 *)
%Y Cf. A000009, A000593, A000629, A008277, A088311, A265024, A305550, A330353, A330388.
%K nonn
%O 1,2
%A _Ilya Gutkovskiy_, Dec 12 2019
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