Rewrite the factorial parts of the summand as

where
is the binomial coefficient, and
are the so-called central binomial coefficients.
Expand the rational expression into partial fractions:

Pull out a constant factor and collect the exponential terms.

The sum we want is now

Let f(x) and g(x) be functions with power series expansions


and recall the well-known binomial series

which converges for |x| < 1/4.
Integrating both sides yields


Taking x = 0 on both sides, it follows that C₁ = -1/2. We then see that

Step back and multiply both sides of the binomial series identity by x, then integrate. This yields


Taking x = 0 again points to C₂ = -1/12. Hence

Then the value of the sum we want is

where ɸ ≈ 1.618 is the golden ratio.