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. That is, the ith row contains only i elements.

Examples
Notable particular examples include these:
 * The Bell triangle, whose numbers count the partitions of a set in which a given element is the largest singleton
 * Catalan's triangle, which counts strings of parentheses in which no close parenthesis is unmatched
 * Euler's triangle, which counts permutations with a given number of ascents
 * Floyd's triangle, whose entries are all of the integers in order
 * Hosoya's triangle, based on the Fibonacci numbers
 * Lozanić's triangle, used in the mathematics of chemical compounds
 * Narayana triangle, counting strings of balanced parentheses with a given number of distinct nestings
 * Pascal's triangle, whose entries are the binomial coefficients

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.

Generalizations
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.

Applications
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.