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

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

Offset

1,2

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

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)

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 *)

Prog

(Magma) [(&*[(&*[i+j: i in [3..j]])/(2*j): j in [3..n+2]]): n in [1..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

Crossrefs

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

Sequence in context: A319831 A122523 A293180 * A219873 A224469 A347907

Adjacent sequences: A203469 A203470 A203471 * A203473 A203474 A203475

Keyword

nonn

Author

Clark Kimberling, Jan 02 2012

Extensions

Name edited by Alois P. Heinz, Jul 23 2017

Status

approved