Narayana Number

Author

Eric W. Weisstein
February 24, 2011

This notebook downloaded from http://mathworld.wolfram.com/notebooks/Combinatorics/NarayanaNumber.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/NarayanaNumber.html.

©2011 Wolfram Research, Inc. except for portions noted otherwise

Formula

𝒩[n_,k_]:=(1/n)Binomial[n,k]Binomial[n,k-1]

FullSimplify[FunctionExpand[𝒩[n,k]],n∈Integers&&k∈Integers]

Triangle

Table[𝒩[n,k],{n,6},{k,n}]//Grid

Sums

Sum[𝒩[n,k],{k,0,n}]

FullSimplify[%-CatalanNumber[n]]

Sum[𝒩[n,k],{k,n}]