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A295407 a(n) = n! * Laguerre(n, 3*n, -n).

%I A295407 #18 Feb 16 2025 08:33:52
%S A295407 1,5,92,2859,124832,7018105,482598720,39236322839,3681751480832,
%T A295407 391611920476653,46560370087846400,6119025385880816035,
%U A295407 880818377346674454528,137824220501484017301281,23291983597732334528110592,4228010378355969165140319375
%N A295407 a(n) = n! * Laguerre(n, 3*n, -n).
%H A295407 G. C. Greubel, <a href="/A295407/b295407.txt">Table of n, a(n) for n = 0..300</a>
%H A295407 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LaguerrePolynomial.html">Laguerre Polynomial</a>
%H A295407 Wikipedia, <a href="https://en.wikipedia.org/wiki/Laguerre_polynomials">Laguerre polynomials</a>
%H A295407 <a href="/index/La#Laguerre">Index entries for sequences related to Laguerre polynomials</a>
%F A295407 a(n) = n!*Sum_{k=0..n} binomial(4*n,n-k)*n^k/k!.
%F A295407 a(n) ~ sqrt(1/2 + 3/(2*sqrt(5))) * (8*(sqrt(5)-1))^n * exp((sqrt(5)-3)*n) * n^n.
%F A295407 a(n) = n! * [x^n] exp(n*x/(1 - x))/(1 - x)^(3*n+1). - _Ilya Gutkovskiy_, Nov 23 2017
%t A295407 Table[n!*LaguerreL[n,3*n,-n],{n,0,15}]
%t A295407 Join[{1},Table[n!*Sum[Binomial[4*n, n-k]*n^k/k!, {k, 0, n}], {n, 1, 15}]]
%o A295407 (PARI) for(n=0,30, print1(n!*sum(k=0, n, binomial(4*n,n-k)*n^k/k!), ", ")) \\ _G. C. Greubel_, Feb 06 2018
%o A295407 (PARI) a(n) = n!*pollaguerre(n, 3*n, -n); \\ _Michel Marcus_, Feb 05 2021
%o A295407 (Magma) [Factorial(n)*(&+[Binomial(4*n,n-k)*n^k/Factorial(k): k in [0..n]]): n in [0..30]]; // _G. C. Greubel_, Feb 06 2018
%Y A295407 Cf. A277373, A295385, A295406, A295408.
%K A295407 nonn
%O A295407 0,2
%A A295407 _Vaclav Kotesovec_, Nov 22 2017