OFFSET
0,2
FORMULA
a(n) = (2*n)! * LaguerreL(2*n, -n).
a(n) = (2*n)! * [x^(2*n)] exp(n*x/(1-x))/(1-x).
a(n) = (2*n)! * Sum_{k=0..2*n} binomial(2*n, k)*n^k / k!.
a(n) ~ 2^(4*n + 1) * n^(2*n) / (sqrt(3) * exp(n)). - Vaclav Kotesovec, May 09 2021
MAPLE
egf := n -> exp(n*x/(1-x))/(1-x): ser := n -> series(egf(n), x, 32):
a := n -> (2*n)!*coeff(ser(n), x, 2*n): seq(a(n), n = 0..13);
MATHEMATICA
a[n_] := HypergeometricU[-2 n, 1, -n];
Table[a[n], {n, 0, 13}]
PROG
(SageMath)
@cached_function
def L(n, x):
if n == 0: return 1
if n == 1: return 1 - x
return (L(n-1, x) * (2*n-1-x) - L(n-2, x)*(n-1))/n
A344049 = lambda n: factorial(2*n)*L(2*n, -n)
print([A344049(n) for n in (0..13)])
(PARI)
a(n) = (2*n)! * sum(j=0, 2*n, binomial(2*n, j) * n^j / j!)
for(n=0, 13, print(a(n)))
CROSSREFS
KEYWORD
nonn
AUTHOR
Peter Luschny, May 08 2021
STATUS
approved