Pascal's triangle: Difference between revisions

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== Formula ==

[[File:PascalTriangleAnimated2.gif|thumb|upright=1|In Pascal's triangle, each number is the sum of the two numbers directly above it.]]In the <math>n</math>th row of Pascal's triangle, the <math>k</math>th entry is denoted <math>\tbinom nk</math>, pronounced "{{mvar|n}} choose {{mvar|k}}". For example, the topmost entry is <math>\tbinom 00 = 1</math>. With this notation, the construction of the previous paragraph may be written as

<math display="block"> {n \choose k} = {n-1 \choose k-1} + {n-1 \choose k}</math>

for any positive integer <math>n</math> and any integer <math>0 \le k \le n</math>.<ref>The binomial coefficient <math>\scriptstyle {n \choose k}</math> is conventionally set to zero if ''k'' is either less than zero or greater than ''n''.</ref> This recurrence for the binomial coefficients is known as [[Pascal's rule]].

== History ==

[[File:Yanghui triangle.gif|thumb|right|upright=1|[[Yang Hui]]'s triangle, as depicted by the Chinese using [[Counting rods|rod numerals]], appears in [[Jade Mirror of the Four Unknowns]], a mathematical work by [[Zhu Shijie]], dated 1303.]]

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Pascal's triangle has many properties and contains many patterns of numbers.

[[File:Pascal's Triangle animated binary rows.gif|thumb|upright=1|Each frame represents a row in Pascal's triangle. Each column of pixels is a number in binary with the least significant bit at the bottom. Light pixels represent 1 and dark pixels 0.]]

[[File:pascal_triangle_compositions.svg|thumb|upright=1|The numbers of [[composition (combinatorics)|compositions]] of ''n''&hairsp;+1 into ''k''&hairsp;+1 ordered partitions form Pascal's triangle.]]

=== Rows ===

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| last = Kallós | first = Gábor

| issue = 1

| journal = Annales Mathématiques Blaise Pascal

| pages = 1–15

| title = A generalization of Pascal's triangle using powers of base numbers

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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.