Moments of combinatorial and Catalan numbers

In this paper we obtain the moments {Φm}m⩾0{Φm}m⩾0 defined byΦm(n):=∑p=1n+1(2p−1)m(2n+1n+1−p)2,n∈N,m∈N0:=N∪{0}, where (mn) is the usual combinatorial number. We also provide the moments in the Catalan triangle whose (n,p)(n,p) entry is defined byAn,p:=2p−12n+1(2n+1n+1−p),n,p∈N,p⩽n+1, and, in particular, new identities involving the well-known Catalan numbers.