A203474 -id:A203474 - cp's OEIS frontend

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

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

Offset: 1

Clark Kimberling, Jan 02 2012

nonn

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.

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

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

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

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

(* 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 *)

[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

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)

Name edited by Alois P. Heinz, Jul 23 2017

A203473 a(n) = v(n+1)/v(n), where v=A203472.

Original entry on oeis.org

7, 72, 990, 17160, 360360, 8910720, 253955520, 8204716800, 296541907200, 11861676288000, 520431047136000, 24858235898496000, 1284342188088960000, 71382386874839040000, 4247252019052922880000

Offset: 1

Clark Kimberling, Jan 02 2012

nonn

G. C. Greubel, Table of n, a(n) for n = 1..250

Cf. A093883, A203472, A203474.

[Floor(Gamma(2*n+6)/Gamma(n+6)): n in [1..16]]; // G. C. Greubel, Aug 27 2023

(* 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}]     (* this sequence *)
Table[v[n]/d[n], {n,20}]        (* A203474 *)
(* Second program *)
Table[Pochhammer[n+6,n], {n,20}] (* G. C. Greubel, Aug 27 2023 *)

[rising_factorial(n+6, n) for n in range(1,16)] # G. C. Greubel, Aug 27 2023

a(n) ~ 2^(2*n + 11/2) * n^n / exp(n). - Vaclav Kotesovec, Apr 09 2021
a(n) = RisingFactorial(6 + n, n). - Peter Luschny, Mar 22 2022
Since v(n) = (135/4)*(2^(n+2)^2/Pi^(n/2))*(BarnesG(n+3)*BarnesG(n+7/2) )/( BarnesG(9/2)*BarnesG(n+6) ) then v(n+1)/v(n) = Gamma(2*n+6) / Gamma(n+6). - G. C. Greubel, Aug 27 2023

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Maple

Mathematica

SageMath

Formula

Extensions