A336100 E.g.f.: Product_{k>=1} (1 - (exp(x) - 1)^k).
1, -1, -3, -7, -15, 89, 1737, 21713, 266865, 3162089, 34737177, 352100033, 2848598145, -7655375911, -1359369828183, -50221626404047, -1460912626424175, -39804558811289911, -1080962878982246343, -29431779044695154527, -788320672341728128095, -20386762121171790275911
Offset: 0
Keywords
Programs
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Mathematica
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 *) 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 *) -
PARI
N=40; x='x+O('x^N); Vec(serlaplace(prod(k=1, N, 1-(exp(x)-1)^k))) -
PARI
f(n) = if( issquare( 24*n + 1, &n), kronecker( 12, n)); \\ A010815 a(n) = sum(k=0, n, stirling(n,k,2) * k! * f(k)); \\ Michel Marcus, Jul 09 2020
Formula
a(n) = Sum_{k=0..n} Stirling2(n,k) * k! * A010815(k).