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;
A356554
Expansion of e.g.f. ( Product_{k>0} 1/(1-x^k)^k )^x.
Original entry on oeis.org
1, 0, 2, 15, 92, 930, 8514, 116760, 1445744, 23020200, 373858920, 6756785640, 130982295432, 2751191997840, 61046788571664, 1445520760702200, 36387213668348160, 960383111961228480, 26780931923301572544, 781864626481646405760, 23925584882896903854720
Offset: 0
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my(N=30, x='x+O('x^N)); Vec(serlaplace(1/prod(k=1, N, (1-x^k)^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, 2)/(j-1)*binomial(i-1, j-1)*v[i-j+1])); v;
A356588
Expansion of e.g.f. ( Product_{k>0} 1/(1 - k * x^k)^(1/k) )^x.
Original entry on oeis.org
1, 0, 2, 9, 44, 450, 2754, 45360, 340304, 6481944, 81801000, 1370631240, 21731534472, 511117017840, 8113055559504, 193958323289640, 4765385232157440, 108183734293844160, 2754467397591689664, 80416694712647352960, 2132862160676063137920, 67803682111729108433280
Offset: 0
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my(N=30, x='x+O('x^N)); Vec(serlaplace(1/prod(k=1, N, (1-k*x^k)^(1/k))^x))
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a055225(n) = sumdiv(n, d, d^(n/d));
a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=2, i, j!*a055225(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 1-4 of 4 results.