A203470 a(n) = Product_{2 <= i < j <= n+1} (i + j).

1, 5, 210, 105840, 838252800, 129459762432000, 466521199899955200000, 45727437650097816797184000000, 139352822480378029387123167068160000000, 14863555768518278744824500982673408262144000000000, 61707340455179609358720715109663452970925870494515200000000000

Offset

1,2

Comments

Each term divides its successor, as (conjectured) in A102693. Each term is divisible by the corresponding superfactorial, A000178(n), as in A203471.

Links

G. C. Greubel, Table of n, a(n) for n = 1..36

Formula

a(n) ~ sqrt(A) * 2^(n^2 + 5*n/2 + 41/24) * exp(-3*n^2/4 + n/2 - 1/24) * n^(n^2/2 - n/2 - 71/24) / Pi, where A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, Apr 08 2021

From G. C. Greubel, Aug 29 2023: (Start)

a(n) = Product_{j=2..n+1} Gamma(2*j)/Gamma(j+2).

a(n) = (2/sqrt(Pi))*( 2^(n+1)^2 * BarnesG(n+5/2)/(Pi^(n/2)*Gamma(n+2)*Gamma(n+3)*BarnesG(3/2)) ).

a(n) = (BarnesG(n+2)/2^n) * Product_{j=2..n+1} Catalan(j). (End)

Maple

a:= n-> mul(mul(i+j, i=2..j-1), j=3..n+1):
seq(a(n), n=1..12);  # Alois P. Heinz, Jul 23 2017

Mathematica

(* First program *)
f[j_]:= j+1; z = 16;
v[n_]:= Product[Product[f[k] + f[j], {j, k-1}], {k, 2, n}]
d[n_]:= Product[(i-1)!, {i, n}]
Table[v[n], {n, z}]           (* A203470 *)
Table[v[n+1]/v[n], {n, z-1}]  (* A102693 *)
Table[v[n]/d[n], {n, 20}]     (* A203471 *)
(* Second program *)
Table[Product[Gamma[2*j]/Gamma[j+2], {j, 2, n+1}], {n, 20}] (* G. C. Greubel, Aug 29 2023 *)

Prog

(Magma) [(&*[Factorial(2*k-1)/Factorial(k+1): k in [2..n+1]]): n in [1..20]]; // G. C. Greubel, Aug 29 2023
(SageMath) [product(gamma(2*k)/gamma(k+2) for k in range(2, n+2)) for n in range(1, 20)] # G. C. Greubel, Aug 29 2023

Crossrefs

Cf. A000178, A102693, A203471.

Sequence in context: A020541 A006413 A055316 * A165208 A293652 A330428

Adjacent sequences: A203467 A203468 A203469 * A203471 A203472 A203473

Keyword

nonn

Author

Clark Kimberling, Jan 02 2012

Extensions

Name edited by Alois P. Heinz, Jul 23 2017

Status

approved