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a(n) = (n!)^n * Sum_{k=1..n} (-1)^(k+1) / k^n.
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%I #8 Jul 14 2020 21:40:01

%S 0,1,3,197,313840,24191662624,137300308036448256,

%T 81994640912971156525105152,6958651785463110878359050928999366656,

%U 108902755985567407887534498777329973193771818418176,395560567918154447056086270973712023435510589158871531520000000000

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

%F a(n) = (n!)^n * [x^n] -polylog(n,-x) / (1 - x).

%t Table[(n!)^n Sum[(-1)^(k + 1)/k^n, {k, 1, n}], {n, 0, 10}]

%t Table[(n!)^n SeriesCoefficient[-PolyLog[n, -x]/(1 - x), {x, 0, n}], {n, 0, 10}]

%o (PARI) a(n) = (n!)^n * sum(k=1, n, (-1)^(k+1) / k^n); \\ _Michel Marcus_, Jul 14 2020

%Y Cf. A024167, A036740, A060943, A142999.

%K nonn

%O 0,3

%A _Ilya Gutkovskiy_, Jul 14 2020