A305987
Expansion of e.g.f. Product_{k>=1} (1 + (exp(x) - 1)^k/k).
Original entry on oeis.org
1, 1, 2, 9, 52, 355, 2976, 29897, 343988, 4423503, 63088600, 992691205, 17095554444, 319404545291, 6427307831504, 138546745515393, 3185841858310180, 77866726065935239, 2016161715005701128, 55127056896177521981, 1587073087715010466556, 47982707153606476112067, 1519931218769637781731712
Offset: 0
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b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
add(combinat[multinomial](n, n-i*j, i$j)/j!*
b(n-i*j, i-1)*(i-1)!^j, j=0..min(1, n/i))))
end:
a:= n-> add(Stirling2(n, j)*b(j$2), j=0..n):
seq(a(n), n=0..25); # Alois P. Heinz, Jun 15 2018
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nmax = 22; CoefficientList[Series[Product[(1 + (Exp[x] - 1)^k/k), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 22; CoefficientList[Series[Exp[Sum[Sum[(-1)^(k + 1) (Exp[x] - 1)^(j k)/(k j^k), {j, 1, nmax}], {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!
b[0] = 1; b[n_] := b[n] = Sum[(n - 1)!/(n - k)! DivisorSum[k, (-#)^(1 - k/#) &] b[n - k], {k, 1, n}]; a[n_] := a[n] = Sum[StirlingS2[n, k] b[k], {k, 0, n}]; Table[a[n], {n, 0, 22}]
A326885
E.g.f.: Product_{k>=1} 1/(1 - k*(exp(x)-1)^k).
Original entry on oeis.org
1, 1, 7, 55, 595, 7351, 110587, 1884415, 36154195, 771983911, 18141124267, 463345240975, 12792709110595, 379854657215671, 12057296962232347, 407072488594360735, 14565548824196479795, 550582832110097346631, 21917855760706255154827, 916261422041320023467695
Offset: 0
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nmax = 20; CoefficientList[Series[Product[1/(1-k*(Exp[x]-1)^k), {k, 1, nmax}], {x, 0, nmax}], x] * Range[0, nmax]!
A326887
E.g.f.: Product_{k>=1} (1 + (exp(x)-1)^k/k) / (1 - (exp(x)-1)^k/k).
Original entry on oeis.org
1, 2, 8, 48, 364, 3320, 35464, 433692, 5962548, 90931152, 1522657264, 27765229844, 547487475484, 11604952395816, 263091290017560, 6351255101776812, 162643987129698628, 4403250400372110656, 125649232950852714496, 3769013390615951560068, 118555772298034094231724
Offset: 0
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nmax = 20; CoefficientList[Series[Product[(1+(Exp[x]-1)^k/k)/(1-(Exp[x]-1)^k/k), {k, 1, nmax}], {x, 0, nmax}], x] * Range[0, nmax]!
A345749
E.g.f.: Product_{k>=1} 1/(1 - (exp(x) - 1)^k)^(1/k).
Original entry on oeis.org
1, 1, 4, 21, 147, 1250, 12633, 147497, 1947676, 28699373, 466994003, 8309274754, 160368858609, 3336869582657, 74468098634660, 1773827462044421, 44905503103938915, 1203843692164105458, 34070243272290551113, 1015056385225183643721
Offset: 0
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max = 19; Range[0, max]! * CoefficientList[Series[Product[1/(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(1/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,numdiv(k)*(exp(x)-1)^k/k))))
A345752
E.g.f.: Product_{k>=1} (1 - (exp(x) - 1)^k / k).
Original entry on oeis.org
1, -1, -2, -3, 0, 55, 572, 4865, 40912, 351675, 2978196, 23418373, 148849544, 84185855, -27459134420, -881482705719, -21652972750464, -487503384038525, -10785437160748156, -242968902040697011, -5627949704687484872, -133358411031825299385
Offset: 0
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max = 21; Range[0, max]! * CoefficientList[Series[Product[1 - (Exp[x] - 1)^k/k, {k, 1, max}], {x, 0, max}], x] (* Amiram Eldar, Jun 26 2021 *)
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my(N=40, x='x+O('x^N)); Vec(serlaplace(prod(k=1, N, 1-(exp(x)-1)^k/k)))
Showing 1-5 of 5 results.
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