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EXAMPLE
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The table of coefficients of x^k in expansion of e.g.f. Product_{k>=1} 1/(1 + x^k)^(n/k) begins:
n = 0: (1), 0, 0, 0, 0, 0, 0, ...
n = 1: 1, (-1), 1, -5, 23, -119, 619, ...
n = 2: 1, -2, (4), -16, 92, -568, 3856, ...
n = 3: 1, -3, 9, (-39), 243, -1737, 13671, ...
n = 4: 1, -4, 16, -80, (536), -4256, 37504, ...
n = 5: 1, -5, 25, -145, 1055, (-9115), 88075, ...
n = 6: 1, -6, 36, -240, 1908, -17784, (185904), ...
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MATHEMATICA
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Table[n! SeriesCoefficient[Product[1/(1 + x^k)^(n/k), {k, 1, n}], {x, 0, n}], {n, 0, 18}]
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