Herbert Wilf has eight Unsolved Problems posted on his webpage. Here, we completely resolve the third of these eight problems. The task seems innocent at first glance: find the first term of the asymptotic behavior of the coefficients of an ordinary generating function. On closer examination, however, the analysis is fraught with difficulties. For instance, the function is the composition of three smaller functions, but the innermost function has a non-zero constant term, so many standard techniques for analyzing function compositions will completely fail. Additionally, the signs of the coefficients are neither all positive, nor alternating in a regular manner. The generating function involves both a square root and an arctangent. The complex-valued square root and arctangent functions each rely on the complex logarithm, which is multivalued and fundamentally depends on a branch cut. These multiple values and branch cuts make the function extremely tedious to visualize using Maple.

The generating function is of interest because the coefficients naturally yield rational approximations to $\pi$.

The proofs rely on complex analysis, in particular, singularity analysis (which, in turn, rely on a Hankel contour and transfer theorems). As a result, we establish an exact formula for the first term of the asymptotic behavior of the coefficients, in the third of Wilf's Unsolved Problems.