BellsTriangle 2007-03-10 18:27:17 2008-06-18 19:47:41 Data Structure Bell number % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here {\em Bell's triangle} or {\em Aitken's array} or {\em 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.