Super Catalan Number
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 
.
The super Catalan numbers are given by the recurrence relation
 |
(1)
|
(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)
|
 |  | ![1/4[-P_n(3)+6P_(n-1)(3)-P_(n-2)(3)]](/images/equations/SuperCatalanNumber/Inline12.gif) |
(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).
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