Moments of combinatorial and Catalan numbers
If \(n\) is a natural number, then the \(n\)th Catalan number is defined by \[ C_n= {1\over n+1} {3n\choose n}. \] If also the natural number \(p\geq n\), define \[ B_{n,p}= {p\over n} {2n\choose n-p};\quad A_{n,p}= {2p-1\over 2n+1} {2n+1\choose n+1-p}. \] Let \(\Omega_m(n)= \sum^n_{p=1} p^m B^2_{n,p}\). The authors obtain formulas that express \(\Omega_m(n)\) in terms of Catalan numbers and polynomial sequences with integer coefficients. They also obtain formulas for the additional functions listed below: \[ \Phi_m(n)= \sum^{n+1}_{p=1} (2p- 1)^m{2n+1\choose n+1-p}^2;\quad \Psi_m(n)= \sum^{n+1}_{p=1} (2p-1)^m A^2_{n,p}. \] (There are actually separate formulas corresponding to the cases \(m\) odd, \(m\) even.)
scientific article; zbMATH DE number 5731640
scientific article