%I #10 Jul 14 2020 21:39:50
%S 1,2,7,172,79745,1375363126,1445639634946657,136511607703654177490168,
%T 1597074319746489837872943936307201,
%U 3049096207067719868011671739966873049880826186,1209808678412717193052533393657339738066086793611743000000001
%N a(n) = (n!)^n * Sum_{k=0..n} 1 / ((k!)^n * (n-k)!).
%F a(n) = (n!)^n * [x^n] exp(x) * Sum_{k>=0} x^k / (k!)^n.
%F a(n) ~ (2*Pi)^((n-1)/2) * n^(n^2 - n/2 + 1/2) / exp(n*(n-1) - 1/12). - _Vaclav Kotesovec_, Jul 14 2020
%t Table[(n!)^n Sum[1/((k!)^n (n - k)!), {k, 0, n}], {n, 0, 10}]
%t Table[(n!)^n SeriesCoefficient[Exp[x] Sum[x^k/(k!)^n, {k, 0, n}], {x, 0, n}], {n, 0, 10}]
%o (PARI) a(n) = (n!)^n * sum(k=0, n, 1 / ((k!)^n * (n-k)!)); \\ _Michel Marcus_, Jul 14 2020
%Y Cf. A000079, A002720, A036740, A119400.
%K nonn
%O 0,2
%A _Ilya Gutkovskiy_, Jul 14 2020