A277423 a(n) = n!*LaguerreL(n, n).

1, 0, -2, 6, 24, -380, 720, 31794, -361088, -2104056, 101548800, -612792290, -25534891008, 593660731404, 2831189530624, -361541172525750, 4481749181890560, 169464194149739536, -6805365045197340672, -9663483091971306186, 6883830206467440640000

Offset: 0

Author

Vaclav Kotesovec, Oct 14 2016

Keywords

• sign

Links

• G. C. Greubel, Table of n, a(n) for n = 0..415
• Eric Weisstein's World of Mathematics, Laguerre Polynomial
• Wikipedia, Laguerre polynomials

Crossrefs

Cf. A277373, A277391, A277392, A277418, A277419, A277420, A277421, A277422.

Cf. A002720, A087912, A277382.

Programs

Magma

[Factorial(n)*(&+[Binomial(n,k)*(-1)^k*n^k/Factorial(k): k in [0..n]]): n in [0..20]]; // G. C. Greubel, May 16 2018

Mathematica

Table[n!*LaguerreL[n, n], {n, 0, 20}]
Flatten[{1, Table[n!*Sum[Binomial[n, k] * (-1)^k * n^k / k!, {k, 0, n}], {n, 1, 20}]}]
Table[n! * Hypergeometric1F1[-n, 1, n], {n, 0, 20}] (* Vaclav Kotesovec, Feb 20 2020 *)

PARI

a(n) = n!*sum(k=0,n, binomial(n,k)*(-1)^k*n^k/k!); \\ G. C. Greubel, May 16 2018

Formula

a(n) = n! * Sum_{k=0..n} binomial(n, k) * (-1)^k * n^k / k!.

a(n) = n! * [x^n] exp(-n*x/(1 - x))/(1 - x). - Ilya Gutkovskiy, Nov 21 2017

a(n) = Sum_{k=0..n} (-n)^(n-k)*k!*binomial(n,k)^2. - Ridouane Oudra, Jul 08 2025