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

%I A317279 #15 Mar 10 2021 03:20:48
%S A317279 1,1,0,-9,-32,225,3456,2695,-433152,-4495743,47872000,1768142871,
%T A317279 6703534080,-597265448351,-11959736205312,126058380654375,
%U A317279 9454322092343296,84694164336894465,-5776865438988238848,-192541299662555831753,1511905067561779200000,243338391925401706938081,3972949090873574466519040
%N A317279 a(n) = Sum_{k=0..n} (-1)^(n-k)*binomial(n-1,k-1)*n^k*n!/k!.
%C A317279 a(n) is the n-th term of the inverse Lah transform of the powers of n.
%H A317279 G. C. Greubel, <a href="/A317279/b317279.txt">Table of n, a(n) for n = 0..400</a>
%H A317279 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%H A317279 <a href="/index/La#Laguerre">Index entries for sequences related to Laguerre polynomials</a>
%F A317279 a(n) = n! * [x^n] exp(n*x/(1 + x)).
%F A317279 a(n) = n! * [x^n] Product_{k>=1} exp(-n*(-x)^k).
%F A317279 a(n) = (-1)^(n+1) * n * n! * Hypergeometric1F1([1-n], [2], n) with a(0) = 1.
%F A317279 a(n) = (-1)^(n+1) * n! * LaguerreL(n-1, 1, n) with a(0) = 1. - _G. C. Greubel_, Mar 09 2021
%p A317279 A317279:= n -> `if`(n=0,1,(-1)^(n+1)*n!*simplify(LaguerreL(n-1,1,n), 'LaguerreL'));
%p A317279 seq(A317279(n), n = 0..30); # _G. C. Greubel_, Mar 09 2021
%t A317279 Join[{1}, Table[Sum[(-1)^(n-k) Binomial[n-1, k-1] n^k n!/k!, {k, n}], {n, 22}]]
%t A317279 Table[n! SeriesCoefficient[Exp[n x/(1 + x)], {x, 0, n}], {n, 0, 22}]
%t A317279 Table[n! SeriesCoefficient[Product[Exp[-n (-x)^k], {k, n}], {x, 0, n}], {n, 0, 22}]
%t A317279 Join[{1}, Table[(-1)^(n+1) n n! Hypergeometric1F1[1-n, 2, n], {n, 22}]]
%o A317279 (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 A317279 (Magma)
%o A317279 l:= func< n, a, b | Evaluate(LaguerrePolynomial(n, a), b) >;
%o A317279 [1]cat[(-1)^(n+1)*Factorial(n)*l(n-1,1,n): n in [1..30]]; // _G. C. Greubel_, Mar 09 2021
%o A317279 (PARI) a(n) = if (n==0, 1, (-1)^(n+1)*n!*pollaguerre(n-1, 1, n)); \\ _Michel Marcus_, Mar 10 2021
%Y A317279 Cf. A111884, A293145, A317277, A317278.
%K A317279 sign
%O A317279 0,4
%A A317279 _Ilya Gutkovskiy_, Jul 25 2018