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A289211 a(n) = n! * Laguerre(n,-5).

%I A289211 #24 Feb 16 2025 08:33:48
%S A289211 1,6,47,446,4929,61870,866695,13373190,224995745,4094022230,
%T A289211 80031878175,1671426609550,37116087808225,872797202471550,
%U A289211 21656891639499575,565266064058561750,15476777687220818625,443409439715399299750,13263588837009155407375
%N A289211 a(n) = n! * Laguerre(n,-5).
%H A289211 Alois P. Heinz, <a href="/A289211/b289211.txt">Table of n, a(n) for n = 0..435</a>
%H A289211 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LaguerrePolynomial.html">Laguerre Polynomial</a>
%H A289211 Wikipedia, <a href="https://en.wikipedia.org/wiki/Laguerre_polynomials">Laguerre polynomials</a>
%H A289211 <a href="/index/La#Laguerre">Index entries for sequences related to Laguerre polynomials</a>
%F A289211 E.g.f.: exp(5*x/(1-x))/(1-x).
%F A289211 a(n) = n! * Sum_{i=0..n} 5^i/i! * binomial(n,i).
%F A289211 a(n) = n! * A160609(n)/A160610(n).
%F A289211 a(n) ~ exp(-5/2 + 2*sqrt(5*n) - n) * n^(n + 1/4) / (sqrt(2)*5^(1/4)) * (1 + 223/(48*sqrt(5*n))). - _Vaclav Kotesovec_, Nov 13 2017
%F A289211 Sum_{n>=0} a(n) * x^n / (n!)^2 = exp(x) * Sum_{n>=0} 5^n * x^n / (n!)^2. - _Ilya Gutkovskiy_, Jul 17 2020
%p A289211 a:= n-> n! * add(binomial(n, i)*5^i/i!, i=0..n):
%p A289211 seq(a(n), n=0..20);
%t A289211 [Table[n!*LaguerreL[n,-5], {n,0,50}]] (* _G. C. Greubel_, May 09 2018 *)
%o A289211 (PARI) for(n=0,30, print1(n!*sum(k=0,n, binomial(n,k)*(5^k/k!)), ", ")) \\ _G. C. Greubel_, May 09 2018
%o A289211 (PARI) a(n) = n!*pollaguerre(n, 0, -5); \\ _Michel Marcus_, Feb 05 2021
%o A289211 (Magma) [(Factorial(n)*(&+[Binomial(n,k)*(5^k/Factorial(k)): k in [0..n]])): n in [0..30]]; // _G. C. Greubel_, May 09 2018
%Y A289211 Column k=5 of A289192.
%Y A289211 Cf. A160609, A160610.
%K A289211 nonn
%O A289211 0,2
%A A289211 _Alois P. Heinz_, Jun 28 2017