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MATHEMATICA
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nmax = 21; CoefficientList[Series[Product[(1 + Log[1 + x]^k)/(1 - Log[1 + x]^k), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 21; CoefficientList[Series[Exp[Sum[(DivisorSigma[1, 2 k] - DivisorSigma[1, k]) Log[1 + x]^k/k, {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 21; CoefficientList[Series[1/EllipticTheta[4, 0, Log[1 + x]], {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[StirlingS1[n, k] Sum[PartitionsQ[j] PartitionsP[k - j], {j, 0, k}] k!, {k, 0, n}], {n, 0, 21}]
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