A combinatorial identity with application to Catalan numbers
| dc.creator | Pan, Hao | |
| dc.creator | Sun, Zhi-Wei | |
| dc.date | 2005-09-27 | |
| dc.date | 2006-10-29 | |
| dc.date.accessioned | 2026-07-07T06:43:08Z | |
| dc.date.available | 2026-07-07T06:43:08Z | |
| dc.description | By a very simple argument, we prove that if $l,m,n$ are nonnegative integers then $$\sum_{k=0}^l(-1)^{m-k}\binom{l}{k}\binom{m-k}{n}\binom{2k}{k-2l+m} =\sum_{k=0}^l\binom{l}{k}\binom{2k}{n}\binom{n-l}{m+n-3k-l}. On the basis of this identity, for $d,r=0,1,2,...$ we construct explicit $F(d,r)$ and $G(d,r)$ such that for any prime $p>\max\{d,r\}$ we have \sum_{k=1}^{p-1}k^r C_{k+d}\equiv \cases F(d,r)(mod p)& if 3|p-1, \\G(d,r)\ (mod p)& if 3|p-2, where $C_n$ denotes the Catalan number $(n+1)^{-1}\binom{2n}{n}$. For example, when $p\geq 5$ is a prime, we have \sum_{k=1}^{p-1}k^2C_k\equiv\cases-2/3 (mod p)& if 3|p-1, \1/3 (mod p)& if 3|p-2; and \sum_{0<k<p-4}\frac{C_{k+4}}k \equiv\cases 503/30 (mod p)& if 3|p-1, -100/3 (mod p)& if 3|p-2. This paper also contains some new recurrence relations for Catalan numbers. | |
| dc.description | 22 pages | |
| dc.identifier | https://arxiv.org/abs/math/0509648 | |
| dc.identifier | http://arxiv.org/abs/math/0509648 | |
| dc.identifier | Discrete Math. 306(2006), 1921-1940 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/102301 | |
| dc.subject | Combinatorics | |
| dc.subject | Number Theory | |
| dc.subject | 11B65; 05A10; 05A19; 11A07; 11B37 | |
| dc.title | A combinatorial identity with application to Catalan numbers | |
| dc.type | text |