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In mathematics and computing, a triangular array of numbers, polynomials, or the like, is a doubly indexed sequence in which each row is only as long as the row's own index.
Notable particular examples include these:
- Bell numbers (the "Bell triangle", "Aitken's array", or the "Peirce triangle")
- Bell polynomials
- Boustrophedon transform
- Eulerian number
- Floyd's triangle
- Hosoya's triangle
- Lah number
- Lozanić's triangle
- Narayana number
- Pascal's triangle
- Rencontres numbers
- Romberg's method
- Stirling numbers of the first kind
- Stirling numbers of the second kind
Triangular arrays of integers in which each row is symmetric and begins and ends with 1 are sometimes called generalized Pascal triangles; examples include Pascal's triangle, the Narayana numbers, and the triangle of Eulerian numbers.
Apart from the representation of triangular matrices, triangular arrays are used in several algorithms. One example is the CKY parsing algorithm for context-free grammars, an example of dynamic programming.
- Triangular number is the number of entries in such an array.
- Barry, P. (2006), "On integer-sequence-based constructions of generalized Pascal triangles", Journal of Integer Sequences 9 (06.2.4): 1–34.
- Hazewinkel, Michiel, ed. (2001), "Arithmetic series", Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4
- Weisstein, Eric W., "Number Triangle", MathWorld.
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