A203516 a(n) = Product_{1 <= i < j <= n} 2*(i+j-1).

1, 4, 192, 184320, 4954521600, 4794391461888000, 204135216112950312960000, 451965950843675288237663846400000, 60040562704967329457107799785403842560000000, 542366306792798635131534558788357929673196306432000000000

Offset: 1

Author

Clark Kimberling, Jan 03 2012

Keywords

• nonn

Comments

Each term divides its successor, as in A034910.

See A093883 for a guide to related sequences.

Links

• G. C. Greubel, Table of n, a(n) for n = 1..33

Crossrefs

Cf. A000178, A005408, A034910, A093883, A203516, A203517.

Programs

Magma

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

Maple

a:= n-> mul(mul(2*(i+j-1), i=1..j-1), j=2..n):
seq(a(n), n=1..12);  # Alois P. Heinz, Jul 23 2017

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), {n, 1, 10}] (* Vaclav Kotesovec, Sep 01 2023 *)

PARI

a(n) = my(pd=1); for(j=1, n, for(i=1, j-1, pd=pd*2*(i+j-1))); pd \\ Felix Fröhlich, Jul 23 2017

SageMath

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

Formula

a(n) ~ sqrt(A) * 2^(-7/24 - n + 3*n^2/2) * exp(-1/24 + n/2 - 3*n^2/4) * n^(1/24 - n/2 + n^2/2), where A = A074962 is the Glaisher-Kinkelin constant. - Vaclav Kotesovec, Sep 01 2023

From G. C. Greubel, Feb 19 2024: (Start)

a(n) = BarnesG(n+1)*A203517(n).

a(n) = 2^binomial(n,2) * Product_{j=1..n-1} (2j)!/j!. (End)

Extensions

Name edited by Alois P. Heinz, Jul 23 2017