A203472 a(n) = Product_{3 <= i < j <= n+2} (i + j).

1, 7, 504, 498960, 8562153600, 3085457671296000, 27493649380770693120000, 6982164025191299372050022400000, 57286678477842677171688269225656320000000, 16987900892972660430046341200043192304533504000000000, 201504981205067832055356568153709798734509139298353152000000000000

Offset: 1

Author

Clark Kimberling, Jan 02 2012

Keywords

• nonn

Comments

Each term divides its successor, as in A203470. It is conjectured that each term is divisible by the corresponding superfactorial, A000178(n), as in A203474.

Links

• G. C. Greubel, Table of n, a(n) for n = 1..35

Crossrefs

Cf. A000178, A093883, A203470, A203473, A203474.

Programs

Magma

[(&*[(&*[i+j: i in [3..j]])/(2*j): j in [3..n+2]]): n in [1..20]]; // G. C. Greubel, Aug 26 2023

Maple

a:= n-> mul(mul(i+j, i=3..j-1), j=4..n+2):
seq(a(n), n=1..12);  # Alois P. Heinz, Jul 23 2017

Mathematica

(* First program *)
f[j_]:= j + 2;    z = 16;
v[n_]:= Product[Product[f[k] + f[j], {j,k-1}], {k,2,n}]
d[n_]:= Product[(i-1)!, {i,n}] (* A000178 *)
Table[v[n], {n,z}]             (* A203472 *)
Table[v[n+1]/v[n], {n,z-1}]    (* A203473 *)
Table[v[n]/d[n], {n,20}]       (* A203474 *)
(* Second program *)
Table[(18*2^(n+2)^2/Pi^(n/2))*BarnesG[n+3]*BarnesG[n+7/2]/(BarnesG[n+ 6]*BarnesG[7/2]), {n,20}] (* G. C. Greubel, Aug 26 2023 *)

SageMath

[product( gamma(2*j)/gamma(j+3) for j in range(3,n+3) ) for n in range(1,20)] # G. C. Greubel, Aug 26 2023

Formula

a(n) ~ 3*sqrt(A) * 2^(n^2 + 9*n/2 + 185/24) * n^(n^2/2 - n/2 - 179/24) / (Pi^(3/2) * exp(3*n^2/4 - n/2 + 1/24)), where A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, Apr 09 2021

From G. C. Greubel, Aug 26 2023: (Start)

a(n) = Prod_{j=3..n+2} Prod_{i=3..j-1} (i + j).

a(n) = Prod_{j=3..n+2} Gamma(2*j)/Gamma(j+3).

a(n) = (18*2^(n+2)^2/Pi^(n/2))*BarnesG(n+3)*BarnesG(n+7/2)/(BarnesG(n+ 6)*BarnesG(7/2)). (End)

Extensions

Name edited by Alois P. Heinz, Jul 23 2017