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A295406 a(n) = n! * Laguerre(n, 2*n, -n).

%I A295406 #20 Feb 16 2025 08:33:52
%S A295406 1,4,58,1422,49000,2174360,118023264,7574532826,561071549056,
%T A295406 47111034709260,4421715905632000,458741213603157254,
%U A295406 52129735913348001792,6439324687323193520608,859089518697047400878080,123108032319553206480143250,18858657171509448248927617024
%N A295406 a(n) = n! * Laguerre(n, 2*n, -n).
%H A295406 G. C. Greubel, <a href="/A295406/b295406.txt">Table of n, a(n) for n = 0..300</a>
%H A295406 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LaguerrePolynomial.html">Laguerre Polynomial</a>
%H A295406 Wikipedia, <a href="https://en.wikipedia.org/wiki/Laguerre_polynomials">Laguerre polynomials</a>
%H A295406 <a href="/index/La#Laguerre">Index entries for sequences related to Laguerre polynomials</a>
%F A295406 a(n) = n!*Sum_{k=0..n} binomial(3*n,n-k)*n^k/k!.
%F A295406 a(n) ~ sqrt(1/2 + 5/(2*sqrt(13))) * ((11 + sqrt(13))/2)^n * exp((sqrt(13)-5)*n/2) * n^n.
%F A295406 a(n) = n! * [x^n] exp(n*x/(1 - x))/(1 - x)^(2*n+1). - _Ilya Gutkovskiy_, Nov 23 2017
%t A295406 Table[n!*LaguerreL[n,2*n,-n],{n,0,15}]
%t A295406 Join[{1},Table[n!*Sum[Binomial[3*n, n-k]*n^k/k!, {k, 0, n}], {n, 1, 15}]]
%o A295406 (PARI) for(n=0,30, print1(n!*sum(k=0, n, binomial(3*n,n-k)*n^k/k!), ", ")) \\ _G. C. Greubel_, Feb 06 2018
%o A295406 (PARI) a(n) = n!*pollaguerre(n, 2*n, -n); \\ _Michel Marcus_, Feb 05 2021
%o A295406 (Magma) [Factorial(n)*(&+[Binomial(3*n,n-k)*n^k/Factorial(k): k in [0..n]]): n in [0..30]]; // _G. C. Greubel_, Feb 06 2018
%Y A295406 Cf. A277373, A295385, A295407, A295408.
%K A295406 nonn
%O A295406 0,2
%A A295406 _Vaclav Kotesovec_, Nov 22 2017