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

%I #9 Mar 08 2018 21:17:37

%S 0,1,5,65,1766,83674,6124584,639826452,90328291248,16558780949136,

%T 3823322392154880,1085461798576638240,371610484248792556800,

%U 150961314165968542273920,71790302154674639506682880,39506878580692178250399571200,24909116615180033772524150937600

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

%F a(n) = n!*n^n*Sum_{k=1..n} 1/(k*n^k).

%e The table of coefficients of x^k in expansion of e.g.f. -log(1 - x)/(1 - n*x) begins:

%e n = 0: (0), 1, 1, 2, 6, 24, ...

%e n = 1: 0, (1), 3, 11, 50, 274, ...

%e n = 2: 0, 1, (5), 32, 262, 2644, ...

%e n = 3: 0, 1, 7, (65), 786, 11814, ...

%e n = 4: 0, 1, 9, 110, (1766), 35344, ...

%e n = 5: 0, 1, 11, 167, 3346, (83674), ...

%e ...

%e This sequence is the main diagonal of the table.

%t Table[n! SeriesCoefficient[-Log[1 - x]/(1 - n x), {x, 0, n}], {n, 0, 16}]

%t Join[{0}, Table[n! n^n Sum[1/(k n^k), {k, 1, n}], {n, 1, 16}]]

%o (PARI) a(n) = n!*n^n*sum(i=1, n, 1/(i*n^i)); \\ _Altug Alkan_, Mar 08 2018

%Y Cf. A000254, A068102, A069015, A104150.

%K nonn

%O 0,3

%A _Ilya Gutkovskiy_, Mar 07 2018