A317279 - OEIS

%I #15 Mar 10 2021 03:20:48

%S 1,1,0,-9,-32,225,3456,2695,-433152,-4495743,47872000,1768142871,

%T 6703534080,-597265448351,-11959736205312,126058380654375,

%U 9454322092343296,84694164336894465,-5776865438988238848,-192541299662555831753,1511905067561779200000,243338391925401706938081,3972949090873574466519040

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

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

%H G. C. Greubel, <a href="/A317279/b317279.txt">Table of n, a(n) for n = 0..400</a>

%H N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>

%H <a href="/index/La#Laguerre">Index entries for sequences related to Laguerre polynomials</a>

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

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

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

%F a(n) = (-1)^(n+1) * n! * LaguerreL(n-1, 1, n) with a(0) = 1. - _G. C. Greubel_, Mar 09 2021

%p A317279:= n -> `if`(n=0,1,(-1)^(n+1)*n!*simplify(LaguerreL(n-1,1,n), 'LaguerreL'));

%p seq(A317279(n), n = 0..30); # _G. C. Greubel_, Mar 09 2021

%t Join[{1}, Table[Sum[(-1)^(n-k) Binomial[n-1, k-1] n^k n!/k!, {k, n}], {n, 22}]]

%t Table[n! SeriesCoefficient[Exp[n x/(1 + x)], {x, 0, n}], {n, 0, 22}]

%t Table[n! SeriesCoefficient[Product[Exp[-n (-x)^k], {k, n}], {x, 0, n}], {n, 0, 22}]

%t Join[{1}, Table[(-1)^(n+1) n n! Hypergeometric1F1[1-n, 2, n], {n, 22}]]

%o (Sage) [1]+[(-1)^(n+1)*factorial(n)*gen_laguerre(n-1,1,n) for n in (1..30)] # _G. C. Greubel_, Mar 09 2021

%o (Magma)

%o l:= func< n, a, b | Evaluate(LaguerrePolynomial(n, a), b) >;

%o [1]cat[(-1)^(n+1)*Factorial(n)*l(n-1,1,n): n in [1..30]]; // _G. C. Greubel_, Mar 09 2021

%o (PARI) a(n) = if (n==0, 1, (-1)^(n+1)*n!*pollaguerre(n-1, 1, n)); \\ _Michel Marcus_, Mar 10 2021

%Y Cf. A111884, A293145, A317277, A317278.

%K sign

%O 0,4

%A _Ilya Gutkovskiy_, Jul 25 2018