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The classical recurrence relation generalizes: the Catalan number of a Coxeter diagram is equal to the sum of the Catalan numbers of all its maximal proper sub-diagrams. [19] The Catalan numbers are a solution of a version of the Hausdorff moment problem. [20]
Substituting k = 1 into this formula gives the Catalan numbers and substituting k = 2 into this formula gives the Schröder–Hipparchus numbers. [7] In connection with the property of Schröder–Hipparchus numbers of counting faces of an associahedron, the number of vertices of the associahedron is given by the Catalan numbers.
Catalan's trapezoids are a countable set of number trapezoids which generalize Catalan’s triangle. Catalan's trapezoid of order m = 1, 2, 3, ... is a number trapezoid whose entries (,) give the number of strings consisting of n X-s and k Y-s such that in every initial segment of the string the number of Y-s does not exceed the number of X-s by m or more. [6]
In mathematics, a recurrence relation is an equation according to which the th term of a sequence of numbers is equal to some combination of the previous terms. Often, only k {\displaystyle k} previous terms of the sequence appear in the equation, for a parameter k {\displaystyle k} that is independent of n {\displaystyle n} ; this number k ...
Recurrence relations (1 C, 31 P) T. Triangles of numbers (29 P) Pages in category "Integer sequences" ... Catalan number; Centered hexagonal number; Collatz conjecture;
Knowing that some Catalan number counts the number of triangulations of a polygon is part of the excitement of what mathematics is all about. Knowing which Catalan number it is, through the mnemonic that "the n-th Catalan number counts the case of n triangles" is intended to be the frosting on the cake. I'd like to put in the count of triangles ...
In mathematics, the n th Motzkin number is the number of different ways of drawing non-intersecting chords between n points on a circle (not necessarily touching every point by a chord). The Motzkin numbers are named after Theodore Motzkin and have diverse applications in geometry , combinatorics and number theory .
Analogously to Pascal's triangle, these numbers may be calculated using the recurrence relation [2] = + (). As base cases, p 1 ( 1 ) = 1 {\displaystyle p_{1}(1)=1} , and any value on the right hand side of the recurrence that would be outside the triangle can be taken as zero.