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EXAMPLE
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The table of coefficients of x^k in expansion of e.g.f. -log(1 - x)/(1 - n*x) begins:
n = 0: (0), 1, 1, 2, 6, 24, ...
n = 1: 0, (1), 3, 11, 50, 274, ...
n = 2: 0, 1, (5), 32, 262, 2644, ...
n = 3: 0, 1, 7, (65), 786, 11814, ...
n = 4: 0, 1, 9, 110, (1766), 35344, ...
n = 5: 0, 1, 11, 167, 3346, (83674), ...
...
This sequence is the main diagonal of the table.
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MATHEMATICA
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Table[n! SeriesCoefficient[-Log[1 - x]/(1 - n x), {x, 0, n}], {n, 0, 16}]
Join[{0}, Table[n! n^n Sum[1/(k n^k), {k, 1, n}], {n, 1, 16}]]
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