  We propose a semi-analytic Stokes expansion ansatz for finite-depth standing
water waves and devise a recursive algorithm to solve the system of
differential equations governing the expansion coefficients. We implement the
algorithm on a supercomputer using arbitrary-precision arithmetic. The Stokes
expansion introduces hyperbolic trigonometric terms that require exponentiation
of power series. We handle this efficiently using Bell polynomials. Under mild
assumptions on the fluid depth, we prove that there are no exact resonances,
though small divisors may occur. Sudden changes in growth rate in the expansion
coefficients are found to correspond to imperfect bifurcations observed when
families of standing waves are computed using a shooting method. A direct
connection between small divisors in the recursive algorithm and imperfect
bifurcations in the solution curves is observed, where the small divisor
excites higher-frequency parasitic standing waves that oscillate on top of the
main wave. A 109th order Pad\'e approximation maintains 25--30 digits of
accuracy on both sides of the first imperfect bifurcation encountered for the
unit-depth problem. This suggests that even if the Stokes expansion is
divergent, there may be a closely related convergent sequence of rational
approximations.
