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A277452 a(n) = Sum_{k=0..n} binomial(n,k) * n^k * k!.

%I A277452 #22 Jan 24 2022 15:54:10
%S A277452 1,2,13,226,7889,458026,39684637,4788052298,766526598721,
%T A277452 157108817646514,40104442275129101,12472587843118746322,
%U A277452 4641978487740740993233,2036813028164774540010266,1040451608604560812273060189,612098707457003526384666111226
%N A277452 a(n) = Sum_{k=0..n} binomial(n,k) * n^k * k!.
%H A277452 Alois P. Heinz, <a href="/A277452/b277452.txt">Table of n, a(n) for n = 0..232</a>
%F A277452 a(n) = exp(1/n) * n^n * Gamma(n+1, 1/n).
%F A277452 a(n) ~ n^n * n!.
%F A277452 a(n) = n! * [x^n] exp(x)/(1 - n*x). - _Ilya Gutkovskiy_, Sep 18 2018
%F A277452 a(n) = floor(n^n*n!*exp(1/n)), n > 0. - _Peter McNair_, Dec 20 2021
%p A277452 a := n -> simplify(hypergeom([1, -n], [], -n)):
%p A277452 seq(a(n), n=0..15); # _Peter Luschny_, Oct 03 2018
%p A277452 # second Maple program:
%p A277452 b:= proc(n, k) option remember;
%p A277452       1 + `if`(n>0, k*n*b(n-1, k), 0)
%p A277452     end:
%p A277452 a:= n-> b(n$2):
%p A277452 seq(a(n), n=0..17);  # _Alois P. Heinz_, May 09 2020
%t A277452 Flatten[{1, Table[Sum[Binomial[n, k]*n^k*k!, {k, 0, n}], {n, 1, 20}]}]
%o A277452 (PARI) a(n) = sum(k=0, n, binomial(n,k) * n^k * k!); \\ _Michel Marcus_, Sep 18 2018
%Y A277452 Main diagonal of A320031.
%Y A277452 Cf. A086331, A277373, A277423, A277453.
%K A277452 nonn
%O A277452 0,2
%A A277452 _Vaclav Kotesovec_, Oct 16 2016