OFFSET
1,2
COMMENTS
LINKS
G. C. Greubel, Table of n, a(n) for n = 1..33
FORMULA
a(n) ~ sqrt(A) * 2^(-7/24 - n + 3*n^2/2) * exp(-1/24 + n/2 - 3*n^2/4) * n^(1/24 - n/2 + n^2/2), where A = A074962 is the Glaisher-Kinkelin constant. - Vaclav Kotesovec, Sep 01 2023
From G. C. Greubel, Feb 19 2024: (Start)
a(n) = BarnesG(n+1)*A203517(n).
a(n) = 2^binomial(n,2) * Product_{j=1..n-1} (2j)!/j!. (End)
MAPLE
a:= n-> mul(mul(2*(i+j-1), i=1..j-1), j=2..n):
seq(a(n), n=1..12); # Alois P. Heinz, Jul 23 2017
MATHEMATICA
f[j_] := 2 j - 1; z = 15;
v[n_] := Product[Product[f[k] + f[j], {j, 1, k - 1}], {k, 2, n}]
d[n_] := Product[(i - 1)!, {i, 1, n}] (* A000178 *)
Table[v[n], {n, 1, z}] (* A203516 *)
Table[v[n + 1]/(4 v[n]), {n, 1, z - 1}] (* A034910 *)
Table[v[n]/d[n], {n, 1, 20}] (* A203517 *)
Table[2^(-1/24 - 3*n/2 + 3*n^2/2) * Glaisher^(3/2) * Pi^(1/4 - n/2) * BarnesG[1/2 + n]/E^(1/8), {n, 1, 10}] (* Vaclav Kotesovec, Sep 01 2023 *)
PROG
(PARI) a(n) = my(pd=1); for(j=1, n, for(i=1, j-1, pd=pd*2*(i+j-1))); pd \\ Felix Fröhlich, Jul 23 2017
(Magma) [2^Binomial(n, 2)*(&*[Factorial(2*k)/Factorial(k): k in [0..n-1]]): n in [1..20]]; // G. C. Greubel, Feb 19 2024
(SageMath) [2^binomial(n, 2)*product(factorial(2*k)/factorial(k) for k in range(n)) for n in range(1, 21)] # G. C. Greubel, Feb 19 2024
CROSSREFS
KEYWORD
nonn
AUTHOR
Clark Kimberling, Jan 03 2012
EXTENSIONS
Name edited by Alois P. Heinz, Jul 23 2017
STATUS
approved