↑ stuart clark:
They are more ways for finding the closed from of your sum.
It is well-known that the following integral representation holds true
After simplifications, one finally deduces that
as required. This concludes the proof.
There is a really nice paper from A.M. Rockett (click here
to download it) devoted to the akin sum
Rockett develops an elemental method using recurrence relations. Generalizing his results, one can obtain similar recurrence relations for the parametric sum
Differentiating the last sum w.r.t. the variable x
, writing 2n
instead of n
and setting x
= -1, one arrives at the sum from the OP.