Sums and polynomials?

Algebra Level 5

Given

$S=\sum_{x=0}^{\infty}\frac {4x \choose x}{2^{4x+1}(3x+1)},$

$$S$$ is the root of a monic third degree polynomial with integral coefficients. Find the sum of coefficients of the polynomial.

This problem would not be possible if not for the awesome problem The Hydra. A good tip would be to try out that problem first.

This problem is part of the set ... and polynomials

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