A203517 a(n) = A203516(n)/A000178(n).

1, 4, 96, 15360, 17203200, 138726604800, 8203736501452800, 3603868630142209228800, 11873738053102139590311936000, 295578185800614925763054760099840000, 55920479534877093093661639943174183976960000

Offset: 1

Author

Clark Kimberling, Jan 03 2012

Keywords

• nonn

Links

• G. C. Greubel, Table of n, a(n) for n = 1..48

Crossrefs

Cf. A000178, A000217, A034910, A093883, A203516.

Programs

Magma

[2^Binomial(n,2)*(&*[Binomial(2*k,k): k in [0..n-1]]): n in [1..20]]; // G. C. Greubel, Feb 19 2024

Mathematica

f[j_] := 2 j - 1; z = 15;
v[n_] := Product[Product[f[k] + f[j], {j, 1, k - 1}], {k, 2, n}]
d[n_] := Product[(i - 1)!, {i, 1, n}]   (* A000178 *)
Table[v[n], {n, 1, z}]                  (* A203516 *)
Table[v[n + 1]/(4 v[n]), {n, 1, z - 1}] (* A034910 *)
Table[v[n]/d[n], {n, 1, 20}]            (* A203517 *)
Table[2^(-1/24 - 3*n/2 + 3*n^2/2) * Glaisher^(3/2) * Pi^(1/4 - n/2) * BarnesG[1/2 + n] / E^(1/8) / BarnesG[1 + n], {n, 1, 12}] (* Vaclav Kotesovec, Sep 01 2023 *)

SageMath

[2^binomial(n,2)*product(binomial(2*k,k) for k in range(n)) for n in range(1,21)] # G. C. Greubel, Feb 19 2024

Formula

a(n) ~ A^(3/2) * 2^(-7/24 - 3*n/2 + 3*n^2/2) * exp(-1/8 + n/2) * n^(1/8 - n/2) / Pi^(n/2), where A = A074962 is the Glaisher-Kinkelin constant. - Vaclav Kotesovec, Sep 01 2023

a(n) = 2^binomial(n,2) * Product_{j=0..n-1} binomial(2*j, j). - G. C. Greubel, Feb 19 2024