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A277372 a(n) = Sum_{k=1..n} binomial(n,n-k)*n^(n-k)*n!/(n-k)!.

%I A277372 #20 Oct 12 2016 08:56:13
%S A277372 0,1,10,141,2584,58745,1602576,51165205,1874935168,77644293201,
%T A277372 3588075308800,183111507687581,10230243235200000,621111794820235849,
%U A277372 40722033570202507264,2867494972696071121125,215840579093024990396416,17294837586403146090259745,1469799445329208661211021312
%N A277372 a(n) = Sum_{k=1..n} binomial(n,n-k)*n^(n-k)*n!/(n-k)!.
%F A277372 a(n) = n!*LaguerreL(n, -n) - n^n.
%F A277372 a(n) = (-1)^n*KummerU(-n, 1, -n) - n^n.
%F A277372 a(n) = n^n*(hypergeom([-n, -n], [], 1/n) - 1) for n>=1.
%F A277372 a(n) ~ n^n * phi^(2*n+1) * exp(n/phi-n) / 5^(1/4), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - _Vaclav Kotesovec_, Oct 12 2016
%p A277372 a := n -> add(binomial(n,n-k)*n^(n-k)*n!/(n-k)!, k=1..n):
%p A277372 seq(a(n), n=0..18);
%p A277372 # Alternatively:
%p A277372 A277372 := n -> n!*LaguerreL(n,-n) - n^n:
%p A277372 seq(simplify(A277372(n)), n=0..18);
%o A277372 (PARI) a(n) = sum(k=1, n, binomial(n,n-k)*n^(n-k)*n!/(n-k)!); \\ _Michel Marcus_, Oct 12 2016
%Y A277372 Cf. A097662, A239768.
%Y A277372 Cf. A002720, A087912, A277382.
%K A277372 nonn
%O A277372 0,3
%A A277372 _Peter Luschny_, Oct 11 2016