Bell triangle

The Bell triangle has been discovered independently by multiple authors, beginning with Charles Sanders Peirce (1880) and including also Alexander Aitken (1933) and Cohn et al. (1962), and for that reason has also been called Aitken's array or the Peirce triangle.

[3] In a format similar to that of Pascal's triangle, and in the order listed in the On-Line Encyclopedia of Integer Sequences (OEIS), its first few rows are:[2] The Bell triangle may be constructed by placing the number 1 in its first position.

An alternative but more technical interpretation of the numbers in the same augmented triangle is given by Quaintance & Kwong (2013).

In this way, as Aitken (1933) observed, this triangle can be interpreted as implementing the Gregory–Newton interpolation formula, which finds the coefficients of a polynomial from the sequence of its values at consecutive integers by using successive differences.

This formula closely resembles a recurrence relation that can be used to define the Bell numbers.