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;
A356565
Expansion of e.g.f. ( Product_{k>0} (1+x^k) )^x.
Original entry on oeis.org
1, 0, 2, 3, 44, 90, 2034, 9240, 168944, 951048, 24042600, 185387400, 4411634952, 44020650960, 1166597641104, 14101322278680, 399099955203840, 5522583764698560, 169123038510919104, 2779010889700890240, 87888034148774728320, 1637061268780618450560
Offset: 0
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nmax = 20; CoefficientList[Series[Product[(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(prod(k=1, N, 1+x^k)^x))
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a000593(n) = sumdiv(n, d, (-1)^(n/d+1)*d);
a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=2, i, j!*a000593(j-1)/(j-1)*binomial(i-1, j-1)*v[i-j+1])); v;
A356566
Expansion of e.g.f. ( Product_{k>0} (1+x^k)^k )^x.
Original entry on oeis.org
1, 0, 2, 9, 92, 510, 7074, 68040, 1002224, 12529944, 228706920, 3565888920, 71035245192, 1348127454960, 30270949077264, 661700017709640, 16516072112482560, 408336559236083520, 11204399270843020224, 309489391954850336640, 9258803420755891835520
Offset: 0
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my(N=30, x='x+O('x^N)); Vec(serlaplace(prod(k=1, N, (1+x^k)^k)^x))
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a078306(n) = sumdiv(n, d, (-1)^(n/d+1)*d^2);
a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=2, i, j!*a078306(j-1)/(j-1)*binomial(i-1, j-1)*v[i-j+1])); v;
Showing 1-3 of 3 results.