Super Catalan Number
While the Catalan Numbers are the number of p-Good Path 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
(Comtet 1974), with
. (Note that the expression in Vardi (1991, p. 198) contains two errors.)A closed form expression in terms of Legendre Polynomials 
is
(Vardi 1991, p. 199). The first few super Catalan numbers are 1, 1, 3, 11, 45, 197, ... (Sloane's A001003).See also Catalan Number
References
Comtet, L. Advanced Combinatorics: The Art of Finite and Infinite Expansions, rev. enl. ed. Dordrecht, Netherlands: Reidel, p. 56, 1974. Graham, R. L.; Knuth, D. E.; and Patashnik, O. Exercise 7.50 in Concrete Mathematics: A Foundation for Computer Science, 2nd ed. Reading, MA: Addison-Wesley, 1994. Motzkin, T. ``Relations Between Hypersurface Cross Ratios and a Combinatorial Formula for Partitions of a Polygon for Permanent Preponderance and for Non-Associative Products.'' Bull. Amer. Math. Soc. 54, 352-360, 1948. Schröder, E. ``Vier combinatorische Probleme.'' Z. Math. Phys. 15, 361-376, 1870. Sloane, N. J. A. SequenceA001003/M2898in ``An On-Line Version of the Encyclopedia of Integer Sequences.''http://www.research.att.com/~njas/sequences/eisonline.html and Sloane, N. J. A. and Plouffe, S.The Encyclopedia of Integer Sequences. San Diego: Academic Press, 1995. Vardi, I. Computational Recreations in Mathematica. Reading, MA: Addison-Wesley, pp. 198-199, 1991.
- Regular Point
- Regular Polygon
- Regular Polyhedron
- Regular Prime
- Regular Ring
- Regular Sequence
- Regular Singularity
- Regular Singular Point
- Regular Surface
- Regular Triangle Center
- Regulus
- Reidemeister Moves
- Reidemeister's Theorem
- Reinhardt Domain
- Relation
- Relative Error
- Relative Extremum
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- Relative Maximum
- Relative Minimum
- Relaxation Methods
- Remainder
- Remainder Theorem
- Rembs' Surfaces
- Removable Crossing