A372986 a(n) = (2*n)!/a(n-1), with a(0)=1.

1, 2, 12, 60, 672, 5400, 88704, 982800, 21288960, 300736800, 8089804800, 138940401600, 4465572249600, 90311261040000, 3375972620697600, 78570797104800000, 3348964839732019200, 88156434351585600000, 4219695698062344192000, 123947946698329353600000

Offset

0,2

Links

Table of n, a(n) for n=0..19.

Formula

a(n) ~ Gamma(1/4) * 2^(n - 1/4) * n^(n + 3/4) / exp(n) if n is even and a(n) ~ sqrt(Pi) * 2^(n + 9/4) * n^(n + 3/4) / (Gamma(1/4) * exp(n)) if n is odd. - Vaclav Kotesovec, Jul 09 2024

a(n) = Product_{k=0..floor((n-1)/2)} 2*(n-2*k)*(2*n-4*k-1). - Andrew Howroyd, Jul 09 2024

Mathematica

a[0] = 1; a[n_] := a[n] = (2 n)!/a[n - 1];
Table[a[n], {n, 0, 30}]
(* or *)
FullSimplify[Table[4^n * Gamma[3/4 + n/2] * Gamma[1 + n/2] / If[EvenQ[n], Sqrt[2]*Pi/Gamma[1/4], Sqrt[Pi]*Gamma[1/4]/4], {n, 0, 20}]] (* Vaclav Kotesovec, Jul 09 2024 *)

Prog

(PARI) a(n)={my(t=1); for(n=1, n, t = (2*n)!/t); t} \\ Andrew Howroyd, Jul 09 2024
(PARI) a(n)={prod(k=0, (n-1)\2, 2*(n-2*k)*(2*n-4*k-1))} \\ Andrew Howroyd, Jul 09 2024

Crossrefs

Cf. A372987, A372988.

Sequence in context: A360590 A362244 A362238 * A190425 A145630 A082688

Adjacent sequences: A372983 A372984 A372985 * A372987 A372988 A372989

Keyword

nonn

Author

Clark Kimberling, Jul 09 2024

Status

approved