A329928 a(n) = (Pi/2)*(2*n+1)!*binomial(2*n+1, (2*n+1)/2).

2, 32, 2048, 294912, 75497472, 30198988800, 17394617548800, 13637380158259200, 13964677282057420800, 18098221757546417356800, 28957154812074267770880000, 56061051716175782404423680000, 129164663154069002659792158720000, 349261249168602583192077997178880000

Offset

0,1

Links

Table of n, a(n) for n=0..13.

Formula

a(n) = 2^(4*n + 1)*Gamma(n + 1)^2.

a(n) = a(n-1)*(4*n)^2 for n > 0.

From Amiram Eldar, Sep 11 2025: (Start)

a(n) ~ Pi * (4*n)^(2*n+1) / exp(2*n).

Sum_{n>=0} 1/a(n) = BesselI(0, 1/2)/2.

Sum_{n>=0} (-1)^n/a(n) = BesselJ(0, 1/2)/2. (End)

Maple

a := proc(n) option remember; if n = 0 then 2 else 16*a(n-1)*n^2 fi end:
seq(a(n), n = 0..13);

Mathematica

a[n_] := (Pi/2) * (2*n+1)! * Binomial[2*n+1, n+1/2]; Array[a, 15, 0] (* Amiram Eldar, Sep 11 2025 *)

Prog

(PARI) \p100; binom(n, k)=gamma(n+1)/(gamma(k+1)*gamma(n-k+1));
for(n=0, 14, print1(round((Pi/2)*(2*n+1)!*binom(2*n+1, (2*n+1)/2)), ", ")) \\ Hugo Pfoertner, Dec 05 2019

Crossrefs

Sequence in context: A087084 A193269 A088386 * A093584 A117259 A234365

Adjacent sequences: A329925 A329926 A329927 * A329929 A329930 A329931

Keyword

nonn

Author

Peter Luschny, Dec 05 2019

Status

approved