Pascal's triangle: Difference between revisions

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| doi = 10.5802/ambp.211

| url = https://ambp.centre-mersenne.org/item/10.5802/ambp.211.pdf

}}.</ref> as demonstrated [[#Binomial expansions|above]]. Thus, when the entries of the row are concatenated and read in radix <math>a</math> they form the numerical equivalent of <math>(a + 1)^{n} = 11^{n}_{a}</math>. If <math>c = a + 1</math> for <math>c < 0</math>, then the theorem [[Negative base|holds]] for <math>a =\bmod 2c</math>, with <math>a</math> congruent to <math>\{c - 1, -(c + 1)\} \;\mathrm{mod}\; 2c</math>, and with odd values of <math>n</math> [[Negative number#Multiplication|yielding]] negative row products.<ref>{{cite book

| display-authors = etal

| last = Hilton | first = P.

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== External links ==

* {{springer|title=Pascal triangle|id=p/p071790}}

* {{MathWorld | urlname=PascalsTriangle | title=PascalsPascal's Triangletriangle }}

* [https://york.ac.uk/depts/maths/histstat/images/triangle.gif The Old Method Chart of the One HundredSeven Multiplying Squares] ''(from the Ssu Yuan Yü Chien of Chu Shi-Chieh, 1303, depicting the first nine rows of Pascal's triangle)''

* [https://web.archive.org/web/20040803130916/https://lib.cam.ac.uk/RareBooks/PascalTraite/ Pascal's Treatise on the Arithmetic Triangle] ''(page images of Pascal's treatise, 1654; [https://web.archive.org/web/20040803233048/https://lib.cam.ac.uk/RareBooks/PascalTraite/pascalintro.pdf summary])''