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

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.