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A340863 a(n) = n!*LaguerreL(n, -n^2).

%I A340863 #33 Feb 16 2025 08:34:01
%S A340863 1,2,34,1626,151064,23046370,5228520912,1651548277946,692979602529664,
%T A340863 372856154213080674,250277853396112428800,205025892171407329263802,
%U A340863 201314381459222197472984064,233396220344077025321595074306
%N A340863 a(n) = n!*LaguerreL(n, -n^2).
%H A340863 Seiichi Manyama, <a href="/A340863/b340863.txt">Table of n, a(n) for n = 0..214</a>
%H A340863 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LaguerrePolynomial.html">Laguerre Polynomial</a>
%H A340863 Wikipedia, <a href="https://en.wikipedia.org/wiki/Laguerre_polynomials">Laguerre polynomials</a>
%H A340863 <a href="/index/La#Laguerre">Index entries for sequences related to Laguerre polynomials</a>
%F A340863 a(n) = Sum_{k=0..n} n^(2*k) * (n-k)! * binomial(n,k)^2.
%F A340863 a(n) = n! * [x^n] exp(n^2 * x/(1-x))/(1-x).
%F A340863 a(n) = A289192(n,n^2).
%F A340863 a(n) ~ exp(1) * n^(2*n). - _Vaclav Kotesovec_, Feb 14 2021
%t A340863 Table[n! * LaguerreL[n, -n^2], {n, 0, 13}] (* _Amiram Eldar_, Feb 05 2021 *)
%o A340863 (PARI) a(n) = sum(k=0, n, n^(2*k)*(n-k)!*binomial(n, k)^2);
%o A340863 (PARI) a(n) = n!*pollaguerre(n, 0, -n^2); \\ _Michel Marcus_, Feb 05 2021
%Y A340863 Main diagonal of A338435.
%Y A340863 Cf. A277373, A289192.
%K A340863 nonn
%O A340863 0,2
%A A340863 _Seiichi Manyama_, Feb 05 2021