A203471 a(n) = v(n)/A000178(n), v = A203470, A000178 = (superfactorials).
1, 5, 105, 8820, 2910600, 3745942200, 18748440711000, 364619674947528000, 27558684271884061296000, 8100324068034882136733280000, 9267305355220395466643896716480000, 41308086890359390753018505224037952000000
Offset: 1
Keywords
Links
- G. C. Greubel, Table of n, a(n) for n = 1..55
Programs
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Magma
[(&*[Factorial(2*k+1)/(Factorial(k-1)*Factorial(k+2)): k in [1..n]]): n in [1..20]]; // G. C. Greubel, Aug 29 2023
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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+2]/(Gamma[j]*Gamma[j+3]), {j,n}], {n,20}] (* G. C. Greubel, Aug 29 2023 *) -
SageMath
[product(gamma(2*k+4)/(gamma(k+1)*gamma(k+4)) for k in range(n)) for n in range(1, 20)] # G. C. Greubel, Aug 29 2023
Formula
From G. C. Greubel, Aug 29 2023: (Start)
a(n) = Product_{j=1..n} Gamma(2*j+2)/(Gamma(j)*Gamma(j+3)).
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)*BarnesG(n+1)) ).
a(n) = (BarnesG(n+2)/(2^n * BarnesG(n+1))) * Product_{j=1..n} Catalan(j+1). (End)
a(n) ~ A^(3/2) * 2^(n^2 + 2*n + 41/24) * exp(n/2 - 1/8) / (n^(n/2 + 23/8) * Pi^(n/2 + 1)), where A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, Nov 19 2023
a(n) = Product_{1 <= j <= i <= n-1} (i + j + 3)/(i - j + 1). - Peter Bala, Oct 25 2024