Definition:Pascal's Triangle

Pascal's Triangle is an array formed by the binomial coefficients:

$$\begin{array}{r|rrrrrrrrrr} n & \binom n 0 & \binom n 1 & \binom n 2 & \binom n 3 & \binom n 4 & \binom n 5 & \binom n 6 & \binom n 7 & \binom n 8 & \binom n 9 \\ [4pt] \hline 0 & 1 & 0 & 0 &  0 &   0 &   0 &  0 &  0 & 0 & 0 \\ 1 & 1 & 1 &  0 &  0 &   0 &   0 &  0 &  0 & 0 & 0 \\ 2 & 1 & 2 &  1 &  0 &   0 &   0 &  0 &  0 & 0 & 0 \\ 3 & 1 & 3 &  3 &  1 &   0 &   0 &  0 &  0 & 0 & 0 \\ 4 & 1 & 4 &  6 &  4 &   1 &   0 &  0 &  0 & 0 & 0 \\ 5 & 1 & 5 & 10 & 10 &   5 &   1 &  0 &  0 & 0 & 0 \\ 6 & 1 & 6 & 15 & 20 &  15 &   6 &  1 &  0 & 0 & 0 \\ 7 & 1 & 7 & 21 & 35 &  35 &  21 &  7 &  1 & 0 & 0 \\ 8 & 1 & 8 & 28 & 56 &  70 &  56 & 28 &  8 & 1 & 0 \\ 9 & 1 & 9 & 36 & 84 & 126 & 126 & 84 & 36 & 9 & 1 \\ \end{array}$$

Also see

 * Pascal's Rule
 * Stirling's Triangles

Historical Notes

 * The earliest reference to it seems to date from between the 5th and 2nd centuries B.C.E. by the Hindu writer Piṅgalá.


 * In Iran it is known as the Khayyam Triangle after Omar Khayyám discussed it in ca. 1100 C.E. It had been discussed even before that by al-Karajī a hundred years previously.


 * In India it was discussed at length by Bhāskara II Āchārya in his ca. 1150 work Līlāvatī.


 * In China it is known as Yang Hui's Triangle after Yang Hui, who himself (in 1261) credited it to Chia Hsien in a work (ca. 1000 C.E.) now lost.


 * It also appears in Chu Shih-Chieh's The Precious Mirror of the Four Elements, published in 1303.


 * The first record of it in Europe seems to be when Petrus Apianus published it on the frontispiece of his 1527 book on business calculations Ein newe und wolgegründete underweisung aller Kauffmanns Rechnung in dreyen Büchern, mit schönen Regeln und fragstücken begriffen.


 * It is also known (particularly in Italy) as Tartaglia's Triangle, after Niccolò Fontana Tartaglia.


 * It was Pascal's 1653 treatise Traité du triangle arithmétique which was perhaps the first time the main properties of this triangle were documented in one place.


 * The name Pascal's Triangle was assigned by Pierre Raymond de Montmort in 1708, and Abraham de Moivre in 1730.