A203513 a(n) = A203312(n+1)/A203312(n).

3, 49, 2028, 159201, 20342448, 3850078401, 1012487793408, 353293863908769, 157973407966483200, 88087149666575064369, 59928191584204259377152, 48860028872008706126041281

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..215

Crossrefs

Cf. A093883, A203312.

Programs

Magma

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

Mathematica

f[j_] := j; z = 12;
v[n_] := Product[Product[f[j]^2 - f[j] f[k] + f[k]^2,
{j, 1, k - 1}], {k, 2, n}]
Table[v[n], {n, 1, z}]          (* A203312 *)
Table[v[n + 1]/v[n], {n, 1, z}] (* A203513 *)
Table[Product[k^2 - k*(n+1) + (n+1)^2, {k, 1, n}], {n, 1, 15}] (* Vaclav Kotesovec, Sep 07 2023 *)

Python

from operator import mul
from functools import reduce
def v(n): return 1 if n==1 else reduce(mul, [reduce(mul, [j**2 - j*k + k**2 for j in range(1, k)]) for k in range(2, n + 1)])
print([v(n + 1)//v(n) for n in range(1, 13)]) # Indranil Ghosh, Jul 26 2017

SageMath

def A203513(n): return product((n+1)*(n-j+1) +j^2 for j in range(1, n+1))
[A203513(n) for n in range(1,31)] # G. C. Greubel, Feb 23 2024

Formula

a(n) ~ exp((n+1)*Pi/(sqrt(3)) - 2*n) * n^(2*n). - Vaclav Kotesovec, Sep 07 2023