Divisibility of generalized factorials

Divisibility of generalized Catalan numbers

We define a q generalization of weighted Catalan numbers studied by Postnikov and Sagan, and prove a result on the divisibility by p of such numbers when p is a prime and q its power.

Some Divisibility Properties of Generalized Fibonacci Sequences

Let c be any square-free integer, p any odd prime such that (c/p) = -1, and n any positive integer. The quantity ./IT, which would ordinarily be defined (mod p) as one of the two solutions of the congruence: x E c (mod p n ) , does not exist. Nevertheless, we may deal with objects of the form a + b/c~(mod p), where a and b are integers, in much the same way that we deal with complex numbers, th...

On a Class of Combinatorial Sums Involving Generalized Factorials

Generalized Catalan Numbers: Linear Recursion and Divisibility

We prove a linear recursion for the generalized Catalan numbers Ca(n) := 1 (a−1)n+1 ( an n ) when a ≥ 2. As a consequence, we show p |Cp(n) if and only if n 6= p−1 p−1 for all integers k ≥ 0. This is a generalization of the well-known result that the usual Catalan number C2(n) is odd if and only if n is a Mersenne number 2 k − 1. Using certain beautiful results of Kummer and Legendre, we give a...

λ-Factorials of n

where Dk is the number of permutations with no fixed points on {1, 2, . . . , k}. In the paper, we utilize the λ-factorials of n, defined by Eriksen, Freij and Wästlund, to give a unified generalization of these two identities. We provide for it a combinatorial proof by the functional digraph theory and two algebraic proofs. Using the umbral representation of our generalized identity and Abel’s...