Question of the value of the following double sum I hope to get the exact value of the following double series:
$$ \sum_{k \geq 0} \sum_{n \geq 0} \binom{2k}{ k} \frac{(-1)^n}{n! (2k+2n+1) 2^{4k+2n+1}}. $$
I am not sure it is possible or not.
I need your comments.

This problem came from the question asked by one of my friends. What is the exact 
value of the following definite integral 
$$ \int_0^{1/2} \frac{exp(-x^2)}{\sqrt{1-x^2}} dx ? $$ 
So, I got the corresponding Taylor series, multiply both of them.
I need closed form for the value, that is, which doesn't have the summation
notation.
