Journal of Integer Sequences, Vol. 8 (2005), Article 05.2.3

A Combinatorial Interpretation of the Numbers 6(2n)!/n!(n+2)!

Ira M. Gessel and Guoce Xin
Department of Mathematics
Brandeis University
Waltham, MA 02454-9110
USA

Abstract: It is well known that the numbers (2m)!(2n)!/m!n!(m+n)! are integers, but in general there is no known combinatorial interpretation for them. When m=0 these numbers are the middle binomial coefficients C(2n,n), and when m=1 they are twice the Catalan numbers. In this paper, we give combinatorial interpretations for these numbers when m=2 or 3.

(Concerned with sequences A007054 and A007272 .)

Received March 5 2005; revised version received March 16 2005. Published in Journal of Integer Sequences April 27 2005.