While the Catalan numbers are the number of p-good paths from to (0,0) which do not cross the diagonal line, the super
Catalan numbers count the number of lattice paths
with diagonal steps from to (0,0) which do not touch the diagonal line .
(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 first few super Catalan numbers are 1, 1, 3, 11, 45, 197, ... (OEIS A001003). These are often called
the "little" Schröder numbers. Multiplying by 2 gives the usual ("large")
Schröder numbers 2, 6, 22, 90, ... (OEIS
A006318).
The first few prime super Catalan numbers have indices 3, 4, 6, 10, 216, ... (OEIS A092839), with no others less than (Weisstein, Mar. 7, 2004), corresponding to the numbers
3, 11, 197, 103049, ... (OEIS A092840).
to (0,0) which do not cross the diagonal line, the super
Catalan numbers count the number of lattice paths
with diagonal steps from 
to (0,0) which do not touch the diagonal line 
.
.
(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/Inline12.svg)
(Weisstein, Mar. 7, 2004), corresponding to the numbers
3, 11, 197, 103049, ... (OEIS A092840).
