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

0, 1, 10, 141, 2584, 58745, 1602576, 51165205, 1874935168, 77644293201, 3588075308800, 183111507687581, 10230243235200000, 621111794820235849, 40722033570202507264, 2867494972696071121125, 215840579093024990396416, 17294837586403146090259745, 1469799445329208661211021312

Offset: 0

Author

Peter Luschny, Oct 11 2016

Keywords

• nonn

Crossrefs

Cf. A097662, A239768.

Cf. A002720, A087912, A277382.

Programs

Maple

a := n -> add(binomial(n,n-k)*n^(n-k)*n!/(n-k)!, k=1..n):
seq(a(n), n=0..18);
# Alternatively:
A277372 := n -> n!*LaguerreL(n,-n) - n^n:
seq(simplify(A277372(n)), n=0..18);

PARI

a(n) = sum(k=1, n, binomial(n,n-k)*n^(n-k)*n!/(n-k)!); \\ Michel Marcus, Oct 12 2016

Formula

a(n) = n!*LaguerreL(n, -n) - n^n.

a(n) = (-1)^n*KummerU(-n, 1, -n) - n^n.

a(n) = n^n*(hypergeom([-n, -n], [], 1/n) - 1) for n>=1.

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