ON THE (n, k)-TH CATALAN NUMBERS

Catalan Numbers Modulo 2 k

In this paper, we develop a systematic tool to calculate the congruences of some combinatorial numbers involving n!. Using this tool, we re-prove Kummer’s and Lucas’ theorems in a unique concept, and classify the congruences of the Catalan numbers cn (mod 64). To achieve the second goal, cn (mod 8) and cn (mod 16) are also classified. Through the approach of these three congruence problems, we ...

The Catalan numbers

E. Catalan stated in 1874 that the numbers (2m)! (2n)!/m! n! (m+n)! are integers. When m = 0 these numbers are the middle binomial coefficients ( 2n n ) . When m = 1 they are twice the Catalan numbers. In this paper, we give a combinatorial interpretation for these numbers when m = 2.

Catalan numbers

Catalan numbers appear in many places throughout mathematics, and are a fairly frequent subject for articles in this magazine. Indeed, a very nice introduction to their properties was given in this magazine by Vun and Belcher [3], and a historical account of their discovery in Chinese mathematics (before Catalan!) has also appeared in a recent article by Larcombe [1]. The article by Vun and Bel...

Approximation of the n-th Root of a Fuzzy Number by Polynomial Form Fuzzy Numbers

In this paper we introduce the root of a fuzzy number, and we present aniterative method to nd it, numerically. We present an algorithm to generatea sequence that can be converged to n-th root of a fuzzy number.

On the Rational Recursive Sequence X_{N+1}=ƔX_{N-K}+(AX_N+BX_{N-K})⁄(CX_N-DX_{N-K})