A335812
E.g.f.: Product_{k>=1} 1 / (1 - (1 - exp(x))^k).
Original entry on oeis.org
1, -1, 3, -7, 39, -31, 1623, 9953, 182199, 2116289, 32269143, 505278113, 9743069559, 214428606209, 5156280298263, 127321200213473, 3176128419544119, 80737907621585729, 2147513299611040983, 61423058495936864033, 1912348969322283717879, 64216042408215934910849
Offset: 0
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nmax = 21; CoefficientList[Series[ Product[1/(1 - (1 - Exp[x])^k), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[(-1)^k StirlingS2[n, k] k! PartitionsP[k], {k, 0, n}], {n, 0, 21}]
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a(n) = sum(k=0, n, (-1)^k * stirling(n,k,2) * k! * numbpart(k)); \\ Michel Marcus, Jun 25 2020
A336100
E.g.f.: Product_{k>=1} (1 - (exp(x) - 1)^k).
Original entry on oeis.org
1, -1, -3, -7, -15, 89, 1737, 21713, 266865, 3162089, 34737177, 352100033, 2848598145, -7655375911, -1359369828183, -50221626404047, -1460912626424175, -39804558811289911, -1080962878982246343, -29431779044695154527, -788320672341728128095, -20386762121171790275911
Offset: 0
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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 *)
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N=40; x='x+O('x^N); Vec(serlaplace(prod(k=1, N, 1-(exp(x)-1)^k)))
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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
A336097
E.g.f.: Product_{k>=1} (1 - (1 - exp(x))^k).
Original entry on oeis.org
1, 1, -1, -5, -13, -149, -1861, -21965, -267373, -3163109, -34739221, -352104125, -3806609533, -67068890069, -1866226978981, -51776974365485, -1180415240484493, -19613026052409029, -122604194898649141, 6950364605049945955, 394565422299921179747, 13840685990526765512011
Offset: 0
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m = 21; Range[0, m]! * CoefficientList[Series[Product[1 - (1 - Exp[x])^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[(-1)^k * StirlingS2[n, k] * k! * A010815[k], {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Jul 09 2020 *)
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N=40; x='x+O('x^N); Vec(serlaplace(prod(k=1, N, 1-(1-exp(x))^k)))
Showing 1-3 of 3 results.
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