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%I #18 Feb 21 2022 03:50:08
%S 1,1,4,30,396,8360,256470,10619952,564959528,37370475648,
%T 3001942868490,287388158562560,32278318416029532,4197544986996581376,
%U 625014083479647028622,105554855135062180485120,20053957030647088382195280,4255329207190209023134564352
%N a(n) = n! * Sum_{k=0..n} (k * (n-k))^k/k!.
%H Vaclav Kotesovec, <a href="/A351795/b351795.txt">Table of n, a(n) for n = 0..266</a>
%H Vaclav Kotesovec, <a href="/A351795/a351795.jpg">Graph - the asymptotic ratio (10000 terms)</a>
%F a(n) ~ sqrt(2*Pi) * n^(2*n + 1/2) / (sqrt(LambertW(exp(2)*n)^2 - 1) * exp(n*(1 - 1/LambertW(exp(2)*n))) * LambertW(exp(2)*n)^n). - _Vaclav Kotesovec_, Feb 20 2022
%t a[n_] := n!*(1 + Sum[(k*(n - k))^k/k!, {k, 1, n}]); Array[a, 18, 0] (* _Amiram Eldar_, Feb 19 2022 *)
%o (PARI) a(n) = n!*sum(k=0, n, (k*(n-k))^k/k!);
%Y Cf. A235328, A349965, A351768, A351796.
%K nonn
%O 0,3
%A _Seiichi Manyama_, Feb 19 2022
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