Umm... Okay, let's do this. The formula that RSolve gave is not very "tight" as far as a good solution is written. It spans about 30 lines. I solved the problem on paper, and though I haven't fully checked the work the expression is giving me the same problem.
Here is a term in the expression:
Product[(Product[2 k - 1, {k, 1, j - 1}]/Product[2 k, {k, 1, j - 1}])^2, {j, 1, n - 1}]
When I try to expand this for large n using Series:
Series[Product[(Product[2 k - 1, {k, 1, j - 1}]/Product[2 k, {k, 1, j - 1}])^2, {j, 1, n - 1}], {n, ∞, 2}]
it simply won't perform the calculation. Again, I'm not faulting the program, I'm asking it to do a rather hard job. I am wondering if there might be a better tool to use for this or if the built in Mathematica tools aren't set up for something like this.
Thanks!
-Dan