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A289214 a(n) = n! * Laguerre(n,-8).

%I A289214 #32 Feb 16 2025 08:33:48
%S A289214 1,9,98,1238,17688,280888,4894672,92676144,1891712384,41361536384,
%T A289214 963532779264,23806296960256,621407739476992,17078348640463872,
%U A289214 492724294444623872,14883442560920164352,469591293625846038528,15443081743064125505536,528340631093887891603456
%N A289214 a(n) = n! * Laguerre(n,-8).
%H A289214 Alois P. Heinz, <a href="/A289214/b289214.txt">Table of n, a(n) for n = 0..431</a>
%H A289214 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LaguerrePolynomial.html">Laguerre Polynomial</a>
%H A289214 Wikipedia, <a href="https://en.wikipedia.org/wiki/Laguerre_polynomials">Laguerre polynomials</a>
%H A289214 <a href="/index/La#Laguerre">Index entries for sequences related to Laguerre polynomials</a>
%F A289214 E.g.f.: exp(8*x/(1-x))/(1-x).
%F A289214 a(n) = n! * Sum_{i=0..n} 8^i/i! * binomial(n,i).
%F A289214 a(n) = n! * A160603(n)/A160604(n).
%F A289214 a(n) ~ exp(-4 + 4*sqrt(2*n) - n) * n^(n + 1/4) / 2^(5/4) * (1 + 451/(96*sqrt(2*n))). - _Vaclav Kotesovec_, Nov 13 2017
%F A289214 Sum_{n>=0} a(n) * x^n / (n!)^2 = exp(x) * Sum_{n>=0} 8^n * x^n / (n!)^2. - _Ilya Gutkovskiy_, Jul 17 2020
%p A289214 a:= n-> n! * add(binomial(n, i)*8^i/i!, i=0..n):
%p A289214 seq(a(n), n=0..20);
%t A289214 Table[n!*LaguerreL[n, -8], {n, 0, 20}] (* _Indranil Ghosh_, Jul 04 2017 *)
%o A289214 (Python)
%o A289214 from mpmath import *
%o A289214 mp.dps=100
%o A289214 def a(n): return int(fac(n)*laguerre(n, 0, -8))
%o A289214 print([a(n) for n in range(21)]) # _Indranil Ghosh_, Jul 04 2017
%o A289214 (PARI) my(x = 'x + O('x^30)); Vec(serlaplace(exp(8*x/(1-x))/(1-x))) \\ _Michel Marcus_, Jul 04 2017
%o A289214 (PARI) a(n) = n!*pollaguerre(n, 0, -8); \\ _Michel Marcus_, Feb 05 2021
%o A289214 (Magma) m:=25; R<x>:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(Exp(8*x/(1-x))/(1-x))); [Factorial(n-1)*b[n]: n in [1..m]]; // _G. C. Greubel_, May 11 2018
%Y A289214 Column k=8 of A289192.
%Y A289214 Cf. A160603, A160604.
%K A289214 nonn
%O A289214 0,2
%A A289214 _Alois P. Heinz_, Jun 28 2017