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E.g.f.: Product_{k>=1} (1 - (exp(x) - 1)^k).
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%I #19 Jul 09 2020 09:40:44

%S 1,-1,-3,-7,-15,89,1737,21713,266865,3162089,34737177,352100033,

%T 2848598145,-7655375911,-1359369828183,-50221626404047,

%U -1460912626424175,-39804558811289911,-1080962878982246343,-29431779044695154527,-788320672341728128095,-20386762121171790275911

%N E.g.f.: Product_{k>=1} (1 - (exp(x) - 1)^k).

%F a(n) = Sum_{k=0..n} Stirling2(n,k) * k! * A010815(k).

%t m = 21; Range[0, m]! * CoefficientList[Series[Product[1 - (Exp[x] - 1)^k, {k, 1, m}], {x, 0, m}], x] (* _Amiram Eldar_, Jul 08 2020 *)

%t A010815[k_] := (m = (1 + Sqrt[1 + 24*k])/6; If[IntegerQ[m], (-1)^m, 0] + If[IntegerQ[m - 1/3], (-1)^(m - 1/3), 0]); Table[Sum[StirlingS2[n, k] * k! * A010815[k], {k, 0, n}], {n, 0, 20}] (* _Vaclav Kotesovec_, Jul 09 2020 *)

%o (PARI) N=40; x='x+O('x^N); Vec(serlaplace(prod(k=1, N, 1-(exp(x)-1)^k)))

%o (PARI) f(n) = if( issquare( 24*n + 1, &n), kronecker( 12, n)); \\ A010815

%o a(n) = sum(k=0, n, stirling(n,k,2) * k! * f(k)); \\ _Michel Marcus_, Jul 09 2020

%Y Cf. A010815, A167137, A335812, A335813, A336097.

%K sign

%O 0,3

%A _Seiichi Manyama_, Jul 08 2020