OFFSET
1,2
LINKS
FORMULA
Integral_{-infinity..infinity} (prod_{j=1..n-1} j^2+x^2)*x/sinh(2*Pi*x) dx = Gamma(n+1/2)^2/(2*n*Pi).
The n-th fraction also equals the n-th coefficient in the expansion of 2F1(1/2,1/2; 1; x) * n!*(n-1)!/2.
EXAMPLE
1/8, 9/64, 75/128, 11025/2048, 178605/2048, 36018675/16384, 2608781175/32768, ...
MATHEMATICA
a[n_] := Gamma[n+1/2]^2/(2*n*Pi) // Numerator; Array[a, 15]
Table[(2*n)!^2 / (n * 2^(4*n+1) * n!^2), {n, 1, 20}] // Numerator (* Vaclav Kotesovec, Mar 11 2015 *)
CROSSREFS
KEYWORD
frac,nonn
AUTHOR
Jean-François Alcover, Mar 11 2015
STATUS
approved