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A344049 a(n) = KummerU(-2*n, 1, -n).

%I A344049 #11 May 09 2021 08:04:29
%S A344049 1,7,648,173007,91356544,80031878175,104921038236672,
%T A344049 192311632290456007,469591293625846038528,1473442955416649975287959,
%U A344049 5776758846811567983984640000,27673221072138317786331655146207,159045755874087839794327707061321728,1080096259061106512089015938295879551727
%N A344049 a(n) = KummerU(-2*n, 1, -n).
%F A344049 a(n) = (2*n)! * LaguerreL(2*n, -n).
%F A344049 a(n) = (2*n)! * [x^(2*n)] exp(n*x/(1-x))/(1-x).
%F A344049 a(n) = (2*n)! * Sum_{k=0..2*n} binomial(2*n, k)*n^k / k!.
%F A344049 a(n) ~ 2^(4*n + 1) * n^(2*n) / (sqrt(3) * exp(n)). - _Vaclav Kotesovec_, May 09 2021
%p A344049 egf := n -> exp(n*x/(1-x))/(1-x): ser := n -> series(egf(n), x, 32):
%p A344049 a := n -> (2*n)!*coeff(ser(n), x, 2*n): seq(a(n), n = 0..13);
%t A344049 a[n_] := HypergeometricU[-2 n, 1, -n];
%t A344049 Table[a[n], {n, 0, 13}]
%o A344049 (SageMath)
%o A344049 @cached_function
%o A344049 def L(n, x):
%o A344049     if n == 0: return 1
%o A344049     if n == 1: return 1 - x
%o A344049     return (L(n-1, x) * (2*n-1-x) - L(n-2, x)*(n-1))/n
%o A344049 A344049 = lambda n: factorial(2*n)*L(2*n, -n)
%o A344049 print([A344049(n) for n in (0..13)])
%o A344049 (PARI)
%o A344049 a(n) = (2*n)! * sum(j=0, 2*n, binomial(2*n, j) * n^j / j!)
%o A344049 for(n=0, 13, print(a(n)))
%Y A344049 a(n) = A344048(2*n, n).
%K A344049 nonn
%O A344049 0,2
%A A344049 _Peter Luschny_, May 08 2021