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

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