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

%I #11 May 09 2021 08:04:29

%S 1,7,648,173007,91356544,80031878175,104921038236672,

%T 192311632290456007,469591293625846038528,1473442955416649975287959,

%U 5776758846811567983984640000,27673221072138317786331655146207,159045755874087839794327707061321728,1080096259061106512089015938295879551727

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

%F a(n) = (2*n)! * LaguerreL(2*n, -n).

%F a(n) = (2*n)! * [x^(2*n)] exp(n*x/(1-x))/(1-x).

%F a(n) = (2*n)! * Sum_{k=0..2*n} binomial(2*n, k)*n^k / k!.

%F a(n) ~ 2^(4*n + 1) * n^(2*n) / (sqrt(3) * exp(n)). - _Vaclav Kotesovec_, May 09 2021

%p egf := n -> exp(n*x/(1-x))/(1-x): ser := n -> series(egf(n), x, 32):

%p a := n -> (2*n)!*coeff(ser(n), x, 2*n): seq(a(n), n = 0..13);

%t a[n_] := HypergeometricU[-2 n, 1, -n];

%t Table[a[n], {n, 0, 13}]

%o (SageMath)

%o @cached_function

%o def L(n, x):

%o if n == 0: return 1

%o if n == 1: return 1 - x

%o return (L(n-1, x) * (2*n-1-x) - L(n-2, x)*(n-1))/n

%o A344049 = lambda n: factorial(2*n)*L(2*n, -n)

%o print([A344049(n) for n in (0..13)])

%o (PARI)

%o a(n) = (2*n)! * sum(j=0, 2*n, binomial(2*n, j) * n^j / j!)

%o for(n=0, 13, print(a(n)))

%Y a(n) = A344048(2*n, n).

%K nonn

%O 0,2

%A _Peter Luschny_, May 08 2021