On divisibility properties of some binomial sums connected with the Catalan and Fibonacci numbers

Abstract

We show that an alternating binomial sum which is connected with the Catalan numbers is divisible by n . A natural generalization of this sum is connected with the generalized Catalan numbers and also divisible by n . A new class of binomial sum is used. In Appendix A, we consider a positive binomial sum connected with Fibonacci and Lucas numbers. In Appendix B, we consider an alternating binomial sum which is also connected with Catalan numbers and divisible by ( a + 1) n + 1. Similar reasoning was already used by the author to reprove more simply Calkin’s result for divisibility of the alternating sum of powers of binomials coefficients by the central binomial coefficient.