A277452 - OEIS

%I #22 Jan 24 2022 15:54:10

%S 1,2,13,226,7889,458026,39684637,4788052298,766526598721,

%T 157108817646514,40104442275129101,12472587843118746322,

%U 4641978487740740993233,2036813028164774540010266,1040451608604560812273060189,612098707457003526384666111226

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

%H Alois P. Heinz, <a href="/A277452/b277452.txt">Table of n, a(n) for n = 0..232</a>

%F a(n) = exp(1/n) * n^n * Gamma(n+1, 1/n).

%F a(n) ~ n^n * n!.

%F a(n) = n! * [x^n] exp(x)/(1 - n*x). - _Ilya Gutkovskiy_, Sep 18 2018

%F a(n) = floor(n^n*n!*exp(1/n)), n > 0. - _Peter McNair_, Dec 20 2021

%p a := n -> simplify(hypergeom([1, -n], [], -n)):

%p seq(a(n), n=0..15); # _Peter Luschny_, Oct 03 2018

%p # second Maple program:

%p b:= proc(n, k) option remember;

%p       1 + `if`(n>0, k*n*b(n-1, k), 0)

%p     end:

%p a:= n-> b(n$2):

%p seq(a(n), n=0..17);  # _Alois P. Heinz_, May 09 2020

%t Flatten[{1, Table[Sum[Binomial[n, k]*n^k*k!, {k, 0, n}], {n, 1, 20}]}]

%o (PARI) a(n) = sum(k=0, n, binomial(n,k) * n^k * k!); \\ _Michel Marcus_, Sep 18 2018

%Y Main diagonal of A320031.

%Y Cf. A086331, A277373, A277423, A277453.

%K nonn

%O 0,2

%A _Vaclav Kotesovec_, Oct 16 2016