# Binomial Coefficient Challenge 5!

Let

$S(m) = \displaystyle \sum_{k=0}^m{\binom{m}{k} \frac{2m+1}{2m+1-k} {(-2)}^k}$

If

$\sum_{m=0}^\infty S(m) 2^{-m} \; =\; \frac{a}{b} \left(1 - \frac{1}{\sqrt{c}}\tanh^{-1}\tfrac{1}{\sqrt{c}}\right),$

where $$a$$ and $$b$$ are positive coprime integers and $$c$$ is not divisible by the square of any prime, then find $$a + b + c$$.

Details and Assumptions

1. You may use a computer to complete the final step but you should know the method to complete the challenge.

2. $$\tanh^{-1}(x) = \frac{1}{2} \ln\left(\frac{1+x}{1-x}\right)$$

×