OFFSET
0,4
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
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
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}]]
PROG
(Sage) [1]+[(-1)^(n+1)*factorial(n)*gen_laguerre(n-1, 1, n) for n in (1..30)] # G. C. Greubel, Mar 09 2021
(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
(PARI) a(n) = if (n==0, 1, (-1)^(n+1)*n!*pollaguerre(n-1, 1, n)); \\ Michel Marcus, Mar 10 2021
CROSSREFS
KEYWORD
sign
AUTHOR
Ilya Gutkovskiy, Jul 25 2018
STATUS
approved