A355064
Expansion of e.g.f. ( Product_{k>0} 1/(1-x^k)^(1/k) )^x.
Original entry on oeis.org
1, 0, 2, 6, 28, 210, 1248, 13020, 102128, 1248912, 13457880, 176726880, 2362784928, 36609693120, 551337892896, 9588702417840, 171779733546240, 3230529997766400, 64714946343904512, 1371420774325866240, 29953522454811096960, 698447624328756610560
Offset: 0
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a[0] := a[0] = 1; a[1] := a[1] = 0;
a[n_] := a[n] = Sum[Factorial[k]*DivisorSigma[0, k - 1]/(k - 1)*Binomial[n - 1, k - 1]* a[n - k], {k, 2, n}];
Table[a[n], {n, 0, 50}] (* Sidney Cadot, Jan 05 2023 *)
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my(N=30, x='x+O('x^N)); Vec(serlaplace(1/prod(k=1, N, (1-x^k)^(1/k))^x))
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a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=2, i, j!*sigma(j-1, 0)/(j-1)*binomial(i-1, j-1)*v[i-j+1])); v;
A354623
Expansion of e.g.f. ( Product_{k>0} 1/(1-x^k) )^x.
Original entry on oeis.org
1, 0, 2, 9, 44, 390, 2754, 32760, 310064, 4244184, 54887400, 818615160, 12909921672, 225872515440, 4045885572624, 79360837887240, 1649832369335040, 35666417240193600, 822291935260976064, 19830352438530840960, 501144432316767688320, 13229590606682042436480
Offset: 0
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nmax = 20; CoefficientList[Series[Product[1/(1 - x^k)^x, {k, 1, nmax}], {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Aug 17 2022 *)
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my(N=30, x='x+O('x^N)); Vec(serlaplace(1/prod(k=1, N, 1-x^k)^x))
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a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=2, i, j!*sigma(j-1)/(j-1)*binomial(i-1, j-1)*v[i-j+1])); v;
A356587
Expansion of e.g.f. ( Product_{k>0} 1/(1 - (k * x)^k)^(1/k) )^x.
Original entry on oeis.org
1, 0, 2, 15, 236, 8490, 459234, 40325880, 4777773104, 767688946920, 156746202491880, 40056474754165320, 12448131138826294152, 4634982982962988690320, 2033625840922821008112144, 1039060311676326627685615800, 611331728108400284878223051520
Offset: 0
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my(N=20, x='x+O('x^N)); Vec(serlaplace(1/prod(k=1, N, (1-(k*x)^k)^(1/k))^x))
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a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=2, i, j!*sigma(j-1, j-1)/(j-1)*binomial(i-1, j-1)*v[i-j+1])); v;
Showing 1-3 of 3 results.