Triangular array

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The triangular array whose right-hand diagonal sequence consists of Bell numbers BellNumberAnimated.gif
The triangular array whose right-hand diagonal sequence consists of Bell numbers

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.

Contents

Examples

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. [9]

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. [10]

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. [11]

Applications

Romberg's method can be used to estimate the value of a definite integral by completing the values in a triangle of numbers. [12]

The Boustrophedon transform uses a triangular array to transform one integer sequence into another. [13]

See also

Related Research Articles

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References

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