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A295409 a(n) = n! * Laguerre(n, n^2, -n).

%I A295409 #27 Feb 16 2025 08:33:52
%S A295409 1,3,58,2859,267576,40818095,9235507968,2906955312471,
%T A295409 1215257338052992,651548571287972859,435901423022852332800,
%U A295409 356000439852418418920643,348583395952381998326141952,403108990190536860168604229031,543577365164816368801494214352896
%N A295409 a(n) = n! * Laguerre(n, n^2, -n).
%H A295409 Robert Israel, <a href="/A295409/b295409.txt">Table of n, a(n) for n = 0..214</a>
%H A295409 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LaguerrePolynomial.html">Laguerre Polynomial</a>
%H A295409 Wikipedia, <a href="https://en.wikipedia.org/wiki/Laguerre_polynomials">Laguerre polynomials</a>
%H A295409 <a href="/index/La#Laguerre">Index entries for sequences related to Laguerre polynomials</a>
%F A295409 a(n) = n! * Sum_{k=0..n} binomial(n*(n+1),n-k)*n^k/k!.
%F A295409 a(n) ~ exp(3/2) * n^(2*n).
%F A295409 a(n) = n! * [x^n] exp(n*x/(1 - x))/(1 - x)^(n^2+1). - _Ilya Gutkovskiy_, Nov 23 2017
%p A295409 seq(n!*orthopoly[L](n,n^2,-n),n=0..30); # _Robert Israel_, Nov 22 2017
%t A295409 Table[n!*LaguerreL[n,n^2,-n],{n,0,15}]
%t A295409 Join[{1},Table[n!*Sum[Binomial[n*(n+1),n-k]*n^k/k!,{k,0,n}],{n,1,15}]]
%o A295409 (PARI) for(n=0,30, print1(n!*sum(k=0,30, binomial(n*(n+1), n-k)*n^k/k!), ", ")) \\ _G. C. Greubel_, May 11 2018
%o A295409 (PARI) a(n) = n!*pollaguerre(n, n^2, -n); \\ _Michel Marcus_, Feb 05 2021
%o A295409 (Magma) [Factorial(n)*(&+[Binomial(n*(n+1), n-k)*n^k/Factorial(k): k in [0..n]]): n in [0..30]]; // _G. C. Greubel_, May 11 2018
%Y A295409 Cf. A277373, A295385, A295406, A295407, A295408, A295418.
%K A295409 nonn
%O A295409 0,2
%A A295409 _Vaclav Kotesovec_, Nov 22 2017