Sum the series: ∑C(2k,k)²/(k*16^k)?

Sum the series: ∑C(2k,k)²/(k*16^k) where C(n,k) is the number of ways to choose k items from among n. The sum goes from k=1 to either n or ∞.





This is a side branch of something I'm working on, and may not be doable. I can get explicit sums for ∑(C(2k,k)/4^k)²/(k+b) where b is a positive integer and the index k starts from 0, but the method breaks down for b=0.

Sum the series: ∑C(2k,k)²/(k*16^k)?
Mathematica is your friend here:


from 1 to infinity, the sum is


4/pi * (pi* log(2) - 2 * c),


where c is Catalan's constant.





I'm pretty sure Mathematica uses the Wilf-Zeilberger algorithm for finding these values, and as such there are other programs that can generate full proofs. Of course, I don't know what these programs are off the top of my head.





Steve