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%I #12 Jun 13 2020 00:48:47
%S 1,1,3,38,1042,49774,3661128,383653080,54275300112,9964363066848,
%T 2303245150868640,654457584668128416,224205104879416320768,
%U 91129285853151907958544,43356207229026959513863680,23868203329368882698589532800,15053662436260897659550535387136
%N a(n) = n! * [x^n] (1 - (n-1)*log(1 + x))/(1 - n*log(1 + x)).
%F a(n) = A317172(n)/n = Sum_{k=0..n} k!*n^(k-1)*Stirling1(n,k) for n > 1.
%F a(n) ~ sqrt(2*Pi) * n^(2*n - 1/2) / exp(n + 1/2). - _Vaclav Kotesovec_, Jun 12 2020
%t a[0] = 1; a[n_] := Sum[k! * n^(k - 1) * StirlingS1[n, k], {k, 0, n}]; Array[a, 17, 0] (* _Amiram Eldar_, Jun 12 2020 *)
%o (PARI) {a(n) = if(n==0, 1, sum(k=0, n, k!*n^(k-1)*stirling(n, k, 1)))}
%Y Main diagonal of A334369.
%Y Cf. A317172, A321189.
%K nonn
%O 0,3
%A _Seiichi Manyama_, Jun 12 2020
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