Catalan's triangle is the number triangle

(OEIS A009766) with entries given by

for nonnegative integers and satisfying , with rows indexed from . With and entries outside the triangle taken as 0, each element in row is equal to the one above plus the one to the left. The sum of row is equal to the last element of the next row and also equal to the Catalan number . Furthermore, .

The coefficients count distinct sequences formed from 1s and s and having nonnegative partial sums. These coefficients were denoted by Bailey (1996), who gave the alternate form

Here the first equation holds for and the second for ; also, for .

See also

Bell Triangle, Clark's Triangle, Euler's Number Triangle, Leibniz Harmonic Triangle, Nonnegative Partial Sum, Number Triangle, Pascal's Triangle, Prime Triangle, Seidel-Entringer-Arnold Triangle

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References

Bailey, D. F. "Counting Arrangements of 1's and 's." Math. Mag. 69, 128-131, 1996.Brualdi, R. A. Introductory Combinatorics, 4th ed. Upper Saddle River, NJ: Prentice Hall, 2004.Forder, H. G. "Some Problems in Combinatorics." Math. Gaz. 45, 199-201, 1961.Shapiro, L. W. "A Catalan Triangle." Disc. Math. 14, 83-90, 1976.Sloane, N. J. A. Sequence A009766 in "The On-Line Encyclopedia of Integer Sequences."

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Catalan's Triangle

Cite this as:

Weisstein, Eric W. "Catalan's Triangle." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/CatalansTriangle.html

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