Bell triangle

 1 1 2 2 3 5 5 7 10 15 15 20 27 37 52 52 67 87 114 151 203
203 255 322 409 523 674 877

The Bell triangle may be constructed by placing the number 1 in its first position. After that placement, the leftmost value in each row of the triangle is filled by copying the rightmost value in the previous row. The remaining positions in each row are filled by a rule very similar to that for Pascal's triangle: they are the sum of the two values to the left and upper left of the position.

Thus, after the initial placement of the number 1 in the top row, it is the last position in its row and is copied to the leftmost position in the next row. The third value in the triangle, 2, is the sum of the two previous values above-left and left of it. As the last value in its row, the 2 is copied into the third row, and the process continues in the same way.

In the same notation, Sun & Wu (2011) augment the triangle with another diagonal to the left of its other values, of the numbers

 1 0 1 1 1 2 1 2 3 5 4 5 7 10 15 11 15 20 27 37 52 41 52 67 87 114 151 203
162 203 255 322 409 523 674 877

This triangle may be constructed similarly to the original version of Bell's triangle, but with a different rule for starting each row: the leftmost value in each row is the difference of the rightmost and leftmost values of the previous row.

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

A different triangle of numbers, with the Bell numbers on only one side, and with each number determined as a weighted sum of nearby numbers in the previous row, was described by Aigner (1999).