Combinatorial Summation!

Calculus Level 5

n=02n+1(2n+1) (2nn)\large{ \sum_{n=0}^{\infty} \dfrac{2^{n+1}}{(2n+1) \ {2n \choose n} }}

If the summation above can be represented as AπB\large{A \cdot \pi^B}, where A,BA,B are positive integers, then find the value of (A+B)3(A+B)^3?

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