A203470 a(n) = Product_{2 <= i < j <= n+1} (i + j).
1, 5, 210, 105840, 838252800, 129459762432000, 466521199899955200000, 45727437650097816797184000000, 139352822480378029387123167068160000000, 14863555768518278744824500982673408262144000000000, 61707340455179609358720715109663452970925870494515200000000000
Offset: 1
Links
- G. C. Greubel, Table of n, a(n) for n = 1..36
Programs
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Magma
[(&*[Factorial(2*k-1)/Factorial(k+1): k in [2..n+1]]): n in [1..20]]; // G. C. Greubel, Aug 29 2023
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Maple
a:= n-> mul(mul(i+j, i=2..j-1), j=3..n+1): seq(a(n), n=1..12); # Alois P. Heinz, Jul 23 2017
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Mathematica
(* First program *) f[j_]:= j+1; z = 16; v[n_]:= Product[Product[f[k] + f[j], {j,k-1}], {k,2,n}] d[n_]:= Product[(i-1)!, {i,n}] Table[v[n], {n, z}] (* A203470 *) Table[v[n+1]/v[n], {n, z-1}] (* A102693 *) Table[v[n]/d[n], {n, 20}] (* A203471 *) (* Second program *) Table[Product[Gamma[2*j]/Gamma[j+2], {j,2,n+1}], {n,20}] (* G. C. Greubel, Aug 29 2023 *) -
Python
from math import prod, factorial def A203470(n): return prod(factorial(2*k+1)//factorial(k+2) for k in range(1,n+1)) # Chai Wah Wu, Aug 26 2025
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SageMath
[product(gamma(2*k)/gamma(k+2) for k in range(2,n+2)) for n in range(1,20)] # G. C. Greubel, Aug 29 2023
Formula
a(n) ~ sqrt(A) * 2^(n^2 + 5*n/2 + 41/24) * exp(-3*n^2/4 + n/2 - 1/24) * n^(n^2/2 - n/2 - 71/24) / Pi, where A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, Apr 08 2021
From G. C. Greubel, Aug 29 2023: (Start)
a(n) = Product_{j=2..n+1} Gamma(2*j)/Gamma(j+2).
a(n) = (2/sqrt(Pi))*( 2^(n+1)^2 * BarnesG(n+5/2)/(Pi^(n/2)*Gamma(n+2)*Gamma(n+3)*BarnesG(3/2)) ).
a(n) = (BarnesG(n+2)/2^n) * Product_{j=2..n+1} Catalan(j). (End)
Extensions
Name edited by Alois P. Heinz, Jul 23 2017
Comments