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A300489 a(n) = n! * [x^n] -log(1 - x)/(1 - n*x).

%I A300489 #9 Mar 08 2018 21:17:37
%S A300489 0,1,5,65,1766,83674,6124584,639826452,90328291248,16558780949136,
%T A300489 3823322392154880,1085461798576638240,371610484248792556800,
%U A300489 150961314165968542273920,71790302154674639506682880,39506878580692178250399571200,24909116615180033772524150937600
%N A300489 a(n) = n! * [x^n] -log(1 - x)/(1 - n*x).
%F A300489 a(n) = n!*n^n*Sum_{k=1..n} 1/(k*n^k).
%e A300489 The table of coefficients of x^k in expansion of e.g.f. -log(1 - x)/(1 - n*x) begins:
%e A300489 n = 0: (0), 1,   1,    2,     6,      24,  ...
%e A300489 n = 1:  0, (1),  3,   11,    50,     274,  ...
%e A300489 n = 2:  0,  1,  (5),  32,   262,    2644,  ...
%e A300489 n = 3:  0,  1,   7,  (65),  786,   11814,  ...
%e A300489 n = 4:  0,  1,   9,  110, (1766),  35344,  ...
%e A300489 n = 5:  0,  1,  11,  167,  3346,  (83674), ...
%e A300489 ...
%e A300489 This sequence is the main diagonal of the table.
%t A300489 Table[n! SeriesCoefficient[-Log[1 - x]/(1 - n x), {x, 0, n}], {n, 0, 16}]
%t A300489 Join[{0}, Table[n! n^n Sum[1/(k n^k), {k, 1, n}], {n, 1, 16}]]
%o A300489 (PARI) a(n) = n!*n^n*sum(i=1, n, 1/(i*n^i)); \\ _Altug Alkan_, Mar 08 2018
%Y A300489 Cf. A000254, A068102, A069015, A104150.
%K A300489 nonn
%O A300489 0,3
%A A300489 _Ilya Gutkovskiy_, Mar 07 2018