A335813
Expansion of e.g.f. Product_{k>=1} (1 + (1 - exp(x))^k).
Original entry on oeis.org
1, -1, 1, -7, -11, -151, -419, -1807, -5291, -381031, -9125939, -139879807, -1217973371, 7055720489, 657464911741, 20268419534993, 455079458957749, 7487596915540409, 62151133224856621, -943454812059725407, -32387452121872219931, 1120264679544729734729
Offset: 0
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nmax = 21; CoefficientList[Series[Product[(1 + (1 - Exp[x])^k), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[(-1)^k StirlingS2[n, k] k! PartitionsQ[k], {k, 0, n}], {n, 0, 21}]
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N=40; x='x+O('x^N); Vec(serlaplace(prod(k=1, N, 1+(1-exp(x))^k))) \\ Seiichi Manyama, Jul 08 2020
A306080
Expansion of e.g.f. Product_{k>=1} (1 + (exp(x) - 1)^k)^k.
Original entry on oeis.org
1, 1, 5, 43, 401, 4651, 64265, 1015603, 17996081, 354373531, 7682286425, 181466541763, 4632985312961, 127068851847211, 3724903637434985, 116185013450349523, 3840969677266089041, 134113334651486325691, 4930511086446971405945, 190327859758408148070883
Offset: 0
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nmax = 20; CoefficientList[Series[Product[(1 + (Exp[x] - 1)^k)^k, {k, 1, nmax}], {x, 0, nmax}], x] * Range[0, nmax]!
A316146
a(n) = Sum_{k=0..n} Stirling2(n,k) * A000009(k) * k^k.
Original entry on oeis.org
1, 5, 67, 865, 15906, 365514, 9545026, 276368635, 9188742238, 343857717788, 13998751394662, 618098575755637, 29469995998980356, 1510585321262760900, 83100039017148288635, 4873627957977247842223, 302388593396139280682588, 19804146883678522219587314
Offset: 1
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Table[Sum[StirlingS2[n, k] * PartitionsQ[k] * k^k, {k, 1, n}], {n, 1, 20}]
A345750
E.g.f.: Product_{k>=1} (1 + (exp(x) - 1)^k)^(1/k).
Original entry on oeis.org
1, 1, 2, 9, 49, 310, 2521, 25557, 290550, 3555041, 48104901, 741103946, 12825399313, 240202011881, 4747281446090, 98808864563065, 2194031697420057, 52582450760730398, 1357237338948268649
Offset: 0
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max = 18; Range[0, max]! * CoefficientList[Series[Product[(1 + (Exp[x] - 1)^k)^(1/k), {k, 1, max}], {x, 0, max}], x] (* Amiram Eldar, Jun 26 2021 *)
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my(N=20, x='x+O('x^N)); Vec(serlaplace(prod(k=1, N, (1+(exp(x)-1)^k)^(1/k))))
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my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(sum(k=1, N, -sumdiv(k, d, (-1)^d)*(exp(x)-1)^k/k))))
A330387
Expansion of e.g.f. Sum_{k>=1} (-1)^(k + 1) * (exp(x) - 1)^k / (k * (1 - (exp(x) - 1)^k)).
Original entry on oeis.org
1, 2, 12, 62, 420, 3782, 40572, 463262, 5708820, 80773622, 1319927532, 23675250062, 447145154820, 8830952572262, 185694817024092, 4246473212654462, 105754322266866420, 2811068529133151702, 78039884046777282252, 2243558766132057764462
Offset: 1
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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
Table[Sum[StirlingS2[n, k] (k - 1)! Sum[Mod[d, 2] d, {d, Divisors[k]}], {k, 1, n}], {n, 1, 20}]
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 *)
A335811
E.g.f.: exp(x) * Product_{k>=1} (1 + (1 - exp(x))^k).
Original entry on oeis.org
1, 0, 0, -6, -36, -270, -1620, -8526, -41076, -549870, -13520340, -262959246, -3587233716, -22581847470, 584571618540, 30096769542834, 859315925548044, 18434866643574930, 285138881159407020, 2045091797042889714, -28367019385288799796, 379914681728984325330
Offset: 0
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nmax = 21; CoefficientList[Series[Exp[x] Product[(1 + (1 - Exp[x])^k), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[(-1)^k StirlingS2[n + 1, k + 1] k! PartitionsQ[k], {k, 0, n}], {n, 0, 21}]
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