# Bell’s triangle

Bell’s triangle or Aitken’s array or Peirce triangle is a triangular arrangement of integers in which the top row has a single 1, and each subsequent row begins with the last number of the previous row, and each remaining number in a row is the sum of the number to the left to the number above left.

The first eight rows are:

 $\begin{array}[]{cccccccccccccccccc}&&&&&&&&&1&&&&&&&&\\ &&&&&&&&1&&2&&&&&&&\\ &&&&&&&2&&3&&5&&&&&&\\ &&&&&&5&&7&&10&&15&&&&&\\ &&&&&15&&20&&27&&37&&52&&&&\\ &&&&52&&67&&87&&114&&151&&203&&&\\ &&&203&&255&&322&&409&&523&&674&&877&&\\ &&877&&1080&&1335&&1657&&2066&&2589&&3263&&4140&\\ &&&&&\vdots&&&&\vdots&&&&\vdots&&&&\\ \end{array}$

These are listed in A011971 of Sloane’s OEIS.

The left outermost diagonal gives the Bell numbers (1, 2, 5, 15, 52, 203, 877, 4140, etc.) in order.

 Title Bell’s triangle Canonical name BellsTriangle Date of creation 2013-03-22 16:49:16 Last modified on 2013-03-22 16:49:16 Owner PrimeFan (13766) Last modified by PrimeFan (13766) Numerical id 5 Author PrimeFan (13766) Entry type Data Structure Classification msc 05A18 Classification msc 11B73 Synonym Bell triangle Synonym Aitken’s array Synonym Aitken array Synonym Peirce triangle Synonym Peirce’s triangle