A317279 a(n) = Sum_{k=0..n} (-1)^(n-k)*binomial(n-1,k-1)*n^k*n!/k!.

1, 1, 0, -9, -32, 225, 3456, 2695, -433152, -4495743, 47872000, 1768142871, 6703534080, -597265448351, -11959736205312, 126058380654375, 9454322092343296, 84694164336894465, -5776865438988238848, -192541299662555831753, 1511905067561779200000, 243338391925401706938081, 3972949090873574466519040

Offset: 0

Author

Ilya Gutkovskiy, Jul 25 2018

Keywords

• sign

Comments

a(n) is the n-th term of the inverse Lah transform of the powers of n.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..400
• N. J. A. Sloane, Transforms
• Index entries for sequences related to Laguerre polynomials

Crossrefs

Cf. A111884, A293145, A317277, A317278.

Programs

Magma

l:= func< n, a, b | Evaluate(LaguerrePolynomial(n, a), b) >;
[1]cat[(-1)^(n+1)*Factorial(n)*l(n-1,1,n): n in [1..30]]; // G. C. Greubel, Mar 09 2021

Maple

A317279:= n -> `if`(n=0,1,(-1)^(n+1)*n!*simplify(LaguerreL(n-1,1,n), 'LaguerreL'));
seq(A317279(n), n = 0..30); # G. C. Greubel, Mar 09 2021

Mathematica

Join[{1}, Table[Sum[(-1)^(n-k) Binomial[n-1, k-1] n^k n!/k!, {k, n}], {n, 22}]]
Table[n! SeriesCoefficient[Exp[n x/(1 + x)], {x, 0, n}], {n, 0, 22}]
Table[n! SeriesCoefficient[Product[Exp[-n (-x)^k], {k, n}], {x, 0, n}], {n, 0, 22}]
Join[{1}, Table[(-1)^(n+1) n n! Hypergeometric1F1[1-n, 2, n], {n, 22}]]

PARI

a(n) = if (n==0, 1, (-1)^(n+1)*n!*pollaguerre(n-1, 1, n)); \\ Michel Marcus, Mar 10 2021

Sage

[1]+[(-1)^(n+1)*factorial(n)*gen_laguerre(n-1,1,n) for n in (1..30)] # G. C. Greubel, Mar 09 2021

Formula

a(n) = n! * [x^n] exp(n*x/(1 + x)).

a(n) = n! * [x^n] Product_{k>=1} exp(-n*(-x)^k).

a(n) = (-1)^(n+1) * n * n! * Hypergeometric1F1([1-n], [2], n) with a(0) = 1.

a(n) = (-1)^(n+1) * n! * LaguerreL(n-1, 1, n) with a(0) = 1. - G. C. Greubel, Mar 09 2021