Wallis-Ramanujan-Schur-Feynman

One of the earliest examples of analytic representations for $\pi$ is given by an infinite product provided by Wallis in 1655. The modern literature often presents this evaluation based on the integral formula $$\frac{2}{\pi} \int_0^\infty \frac{dx}{(x^2+1)^{n+1}} = \frac{1}{2^{2n}} \binom{2n}{n}.$$ In trying to understand the behavior of this integral when the integrand is replaced by the inverse of a product of distinct quadratic factors, the authors encounter relations to some formulas of Ramanujan, expressions involving Schur functions, and Matsubara sums that have appeared in the context of Feynman diagrams.