Few More Series of Reciprocals with Binomial Coefficients and Their Evaluations

In the present work, utilizing the known series, new series involving reciprocals of binomial coefficients, alternating positive, and negative binomial coefficients are constructed. Further, several new series of reciprocals of binomial coefficients with two odd terms in the denominator are obtained. In the end, we use these to establish the closed form evaluations of hypergeometric functions for the argument 1/16.