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MATHEMATICA
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nmax = 19; CoefficientList[Series[Product[(1 + Log[1/(1 - x)]^k)/(1 - Log[1/(1 - x)]^k), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 19; CoefficientList[Series[Exp[Sum[(DivisorSigma[1, 2 k] - DivisorSigma[1, k]) Log[1/(1 - x)]^k/k, {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 19; CoefficientList[Series[1/EllipticTheta[4, 0, Log[1/(1 - x)]], {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[Abs[StirlingS1[n, k]] Sum[PartitionsQ[j] PartitionsP[k - j], {j, 0, k}] k!, {k, 0, n}], {n, 0, 19}]
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