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%I #16 Feb 19 2022 04:42:30
%S 1,1,12,279,11536,746525,69768036,8902181575,1487939919936,
%T 315597946293657,82839437215344100,26366747854082944451,
%U 10006618140321691249296,4464690010732922712332149,2313871692128866349730705924,1378552938661073773617331110975
%N a(n) = n! * Sum_{k=0..n} n^(n-k) * (n-k)^k/k!.
%F a(n) = n! * [x^n] 1/(1 - n*x*exp(x)).
%F From _Vaclav Kotesovec_, Feb 19 2022: (Start)
%F a(n) ~ exp(1) * n! * n^n.
%F a(n) ~ sqrt(2*Pi) * n^(2*n + 1/2) / exp(n-1). (End)
%t Join[{1}, Table[n!*Sum[n^(n - k)*(n - k)^k/k!, {k, 0, n}], {n, 1, 20}]] (* _Vaclav Kotesovec_, Feb 19 2022 *)
%o (PARI) a(n) = n!*sum(k=0, n, n^(n-k)*(n-k)^k/k!);
%Y Main diagonal of A351761.
%Y Cf. A332408, A351768, A351779.
%K nonn
%O 0,3
%A _Seiichi Manyama_, Feb 18 2022
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