“Super Catalan Number”的概念、定义、习题解题方法及翻译-数学辞典

单词

Super Catalan Number

释义

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

(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.

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