A203310 a(n) = A203309(n+1)/A203309(n).

1, 2, 15, 252, 7560, 356400, 24324300, 2270268000, 277880803200, 43197833952000, 8315583035760000, 1942008468966720000, 540988073497872000000, 177227692877902867200000, 67457290601651778828000000, 29522484828017013792960000000, 14721879100904484211422720000000

Offset: 0

Author

Clark Kimberling, Jan 01 2012

Keywords

• nonn

Links

• G. C. Greubel, Table of n, a(n) for n = 0..200

Crossrefs

Cf. A203306, A203309.

Programs

Magma

F:= Factorial; [(F(n)*F(2*n+2))/(2^n*F(n+2)): n in [0..20]]; // G. C. Greubel, Aug 29 2023

Maple

b:= proc(n) option remember; uses LinearAlgebra;
      Determinant(VandermondeMatrix([i*(i+1)/2$i=1..n]))
    end:
a:= n-> b(n+1)/b(n):
seq(a(n), n=0..16);  # Alois P. Heinz, Aug 29 2023

Mathematica

(* First program *)
f[j_]:= j*(j+1)/2; z = 15;
v[n_]:= Product[Product[f[k] - f[j], {j,k-1}], {k,2,n}]
Table[v[n], {n,z}]             (* A203309 *)
Table[v[n+1]/v[n], {n,0,z-1}]  (* A203310 *)
(* Second program *)
Table[(n!*(2*n+2)!)/(2^n*(n+2)!), {n,0,20}] (* G. C. Greubel, Aug 29 2023 *)

Python

from operator import mul
from functools import reduce
def f(n): return n*(n + 1)//2
def v(n): return 1 if n==1 else reduce(mul, (f(k) - f(j) for k in range(2, n + 1) for j in range(1, k)))
print([v(n + 1)//v(n) for n in range(1, 15)]) # Indranil Ghosh, Jul 24 2017

SageMath

f=factorial; [(f(n)*f(2*n+2))/(2^n*f(n+2)) for n in range(21)] # G. C. Greubel, Aug 29 2023

Formula

a(n) ~ sqrt(Pi) * 2^(n+3) * n^(2*n + 1/2) / exp(2*n). - Vaclav Kotesovec, Jan 25 2019

a(n) = (n!*(2*n+2)!)/(2^n*(n+2)!). - G. C. Greubel, Aug 29 2023

Extensions

Name corrected by Vaclav Kotesovec, Jan 25 2019

a(0)=1 prepended by Alois P. Heinz, Aug 29 2023