A203515 a(n) = A203514(n+1)/A203514(n).

13, 1519, 490827, 310285521, 323965491213, 505036803636351, 1099306007175141675, 3185114376029382371169, 11851908573273735083748813, 55083172732097477388836049999, 312715835695576039538837531922507

Offset: 1

Author

Clark Kimberling, Jan 04 2012

Keywords

• nonn

Comments

See A093883 for a discussion and guide to related sequences.

Links

• G. C. Greubel, Table of n, a(n) for n = 1..185

Crossrefs

Cf. A093883, A203514.

Programs

Magma

[(&*[(2*n+1)^3 -(2*j-1)^3: j in [1..n]])/(2^n*Factorial(n)): n in [1..30]]; // G. C. Greubel, Feb 23 2024

Mathematica

(* First program *)
f[j_]:= 2 j - 1; z = 12;
v[n_]:= Product[f[j]^2 + f[j]*f[k] + f[k]^2, {k,2,n}, {j,k-1}]
Table[v[n], {n, z}]           (* A203514 *)
Table[v[n + 1]/v[n], {n, z}]  (* A203515 *)
(* Second program *)
A203515[n_]:= Product[(2*n+1)^3 - (2*j-1)^3, {j,n}]/(2^n*n!);
Table[A203515[n], {n,30}] (* G. C. Greubel, Feb 23 2024 *)

SageMath

def A203515(n): return product((2*n+1)^3 -(2*j-1)^3 for j in range(1, n+1))/(2^n*factorial(n))
[A203515(n) for n in range(1,31)] # G. C. Greubel, Feb 23 2024

Formula

a(n) = (1/(2^n * n!))*Product_{j=1..n} ((2*n+1)^3 - (2*j-1)^3). - G. C. Greubel, Feb 23 2024