The super Catalan number , for a positive
integer, counts the lattice paths
from
to (0,0) using steps ,, and that stay on or below the diagonal and have no diagonal step lying on it.
for
(Comtet 1974), with . (Note that the expression
in Vardi (1991, p. 198) contains two errors.) A closed form expression
in terms of Legendre polynomials for is
(2)
(3)
(Vardi 1991, p. 199). The values for , 2, ... are 1, 1, 3, 11, 45,
197, ... (OEIS A001003). These are often called
the "little" Schröder numbers. The usual ("large") Schröder
numbers
satisfy
for , while . The values for , 2, ... are 2, 6, 22, 90, ... (OEIS A006318).
The first few prime super Catalan numbers have indices 3, 4, 6, 10, 216, ... (OEIS A092839), corresponding
to the numbers 3, 11, 197, 103049, ... (OEIS A092840).
E. W. Weisstein reported no other such indices through on Mar. 16, 2004.
, for a positive
integer 
, counts the lattice paths
from 
to (0,0) using steps 
, 
, and 
that stay on or below the diagonal 
and have no diagonal step lying on it.
(Comtet 1974), with 
. (Note that the expression
in Vardi (1991, p. 198) contains two errors.) A closed form expression
in terms of Legendre polynomials 
for 
is
![1/4[-P_n(3)+6P_(n-1)(3)-P_(n-2)(3)]](/images/equations/SuperCatalanNumber/Inline17.svg)
, 2, ... are 1, 1, 3, 11, 45,
197, ... (OEIS A001003). These are often called
the "little" Schröder numbers. The usual ("large") Schröder
numbers 
satisfy 
for 
, while 
. The values for 
, 2, ... are 2, 6, 22, 90, ... (OEIS A006318).
on Mar. 16, 2004.
