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MATHEMATICA
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nmax = 20; CoefficientList[Series[Product[Exp[x^k] * (1 + x^k), {k, 1, nmax}], {x, 0, nmax}], x] * Range[0, nmax]!
nmax = 20; CoefficientList[Series[Exp[x/(1 - x)] * Product[(1 + x^k), {k, 1, nmax}], {x, 0, nmax}], x] * Range[0, nmax]!
nmax = 20; CoefficientList[Series[Product[(1 + x^k) / (1 - x^k)^(EulerPhi[k]/k), {k, 1, nmax}], {x, 0, nmax}], x] * Range[0, nmax]!
Table[n!*Sum[LaguerreL[k, -1, -1]*PartitionsQ[n-k], {k, 0, n}], {n, 0, 20}]
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