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%I #6 Mar 08 2018 21:17:29
%S 0,1,7,65,770,11149,191124,3788469,85281552,2149582761,59983774240,
%T 1835925702137,61157508893568,2202760340194517,85303050939131648,
%U 3534478528925155725,156026612737389987840,7310587974761946511761,362356607517279564386304,18943214212273585171456753
%N a(n) = n! * [x^n] -exp(n*x)*log(1 - x)/(1 - x).
%H N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%F a(n) = Sum_{k=1..n} n^(n-k)*binomial(n,k)*k!*H(k), where H(k) is the k-th harmonic number.
%e The table of coefficients of x^k in expansion of e.g.f. -exp(n*x)*log(1 - x)/(1 - x) begins:
%e n = 0: (0), 1, 3, 11, 50, 274, ...
%e n = 1: 0, (1), 5, 23, 116, 669, ...
%e n = 2: 0, 1, (7), 41, 242, 1534, ...
%e n = 3: 0, 1, 9, (65), 452, 3229, ...
%e n = 4: 0, 1, 11, 95, (770), 6234, ...
%e n = 5: 0, 1, 13, 131, 1220, (11149), ...
%e ...
%e This sequence is the main diagonal of the table.
%t Table[n! SeriesCoefficient[-Exp[n x] Log[1 - x]/(1 - x), {x, 0, n}], {n, 0, 19}]
%t Table[Sum[n^(n - k) Binomial[n,k] k! HarmonicNumber[k], {k, 1, n}], {n, 0, 19}]
%Y Cf. A000254, A065456, A073596.
%K nonn
%O 0,3
%A _Ilya Gutkovskiy_, Mar 07 2018