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%I #6 Jul 22 2020 10:28:38
%S 0,1,3,107,109720,5916402624,25690641168448256,
%T 12501662072725447325457536,901886074956174349048867091963183104,
%U 12343856662712388173832816538241443833756015132672,39989244654801819205752864236178211163455535276138236680981184512
%N a(n) = (n!)^n * [x^n] -log(1 - Sum_{k>=1} x^k / k^n).
%t Table[(n!)^n SeriesCoefficient[-Log[1 - Sum[x^k/k^n, {k, 1, n}]], {x, 0, n}], {n, 0, 10}]
%t b[n_, k_] := If[n == 0, 0, ((n - 1)!)^k + (1/n) Sum[(Binomial[n, j] (n - j - 1)!)^k j b[j, k], {j, 1, n - 1}]]; a[n_] := b[n, n]; Table[a[n], {n, 0, 10}]
%Y Cf. A003713, A074708, A336436, A336437, A336439, A336440.
%K nonn
%O 0,3
%A _Ilya Gutkovskiy_, Jul 21 2020
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