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;
A356564
Expansion of e.g.f. ( Product_{k>0} (1+x^k)^(1/k) )^x.
Original entry on oeis.org
1, 0, 2, 0, 28, -30, 888, -1260, 51728, -196560, 5293080, -22286880, 710229408, -4851269280, 138348035616, -1091188098000, 36482139114240, -379928382462720, 11812558481332992, -137793570801143040, 4609972759421554560, -67292912045817561600
Offset: 0
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my(N=30, x='x+O('x^N)); Vec(serlaplace(prod(k=1, N, (1+x^k)^(1/k))^x))
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a048272(n) = sumdiv(n, d, (-1)^(n/d+1));
a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=2, i, j!*a048272(j-1)/(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;
Showing 1-3 of 3 results.