Rational combinations for the sums involving inverse binomial coefficients

We say that the number r can be expressed as a rational combination of the set T = {fi∣i ∈ I} if there exist rational numbers di such that r=∑i∈Idifi. In this paper, we study rational combinations for sums involving inverse binomial coefficients of the form:∑n⩾0npn+kqn-1xn,(x∈Q)where p,q,k∈N0 and {an} is any rational sequence. In several interesting cases of the sum, we obtain closed forms, recursion formulas and experimental results.