Triangular array

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:

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.

Triangular arrays may list mathematical values other than numbers; for instance the Bell polynomials form a triangular array in which each array entry is a polynomial.

Arrays in which the length of each row grows as a linear function of the row number (rather than being equal to the row number) have also been considered.

Apart from the representation of triangular matrices, triangular arrays are used in several algorithms. One example is the CYK algorithm for parsing context-free grammars, an example of dynamic programming.

Romberg's method can be used to estimate the value of a definite integral by completing the values in a triangle of numbers.

The Boustrophedon transform uses a triangular array to transform one integer sequence into another.

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