  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.
