# Definition:Pascal's Triangle

## Definition

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

- $\begin{array}{r|rrrrrrrrrr} n & \binom n 0 & \binom n 1 & \binom n 2 & \binom n 3 & \binom n 4 & \binom n 5 & \binom n 6 & \binom n 7 & \binom n 8 & \binom n 9 & \binom n {10} & \binom n {11} & \binom n {12} \\ \hline 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 2 & 1 & 2 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 3 & 1 & 3 & 3 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 4 & 1 & 4 & 6 & 4 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 5 & 1 & 5 & 10 & 10 & 5 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 6 & 1 & 6 & 15 & 20 & 15 & 6 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 7 & 1 & 7 & 21 & 35 & 35 & 21 & 7 & 1 & 0 & 0 & 0 & 0 & 0 \\ 8 & 1 & 8 & 28 & 56 & 70 & 56 & 28 & 8 & 1 & 0 & 0 & 0 & 0 \\ 9 & 1 & 9 & 36 & 84 & 126 & 126 & 84 & 36 & 9 & 1 & 0 & 0 & 0 \\ 10 & 1 & 10 & 45 & 120 & 210 & 252 & 210 & 120 & 45 & 10 & 1 & 0 & 0 \\ 11 & 1 & 11 & 55 & 165 & 330 & 462 & 462 & 330 & 165 & 55 & 11 & 1 & 0 \\ 12 & 1 & 12 & 66 & 220 & 495 & 792 & 924 & 792 & 495 & 220 & 66 & 12 & 1 \\ \end{array}$

This sequence is A007318 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).

### Row

Each of the horizontal lines of numbers corresponding to a given $n$ is known as the **$n$th row** of Pascal's triangle.

Hence the top **row**, containing a single $1$, is identified as the **zeroth row**, or **row $0$**.

### Column

Each of the vertical lines of numbers headed by a given $\dbinom n m$ is known as the **$m$th column** of Pascal's triangle.

Hence the leftmost **column**, containing all $1$s, is identified as the **zeroth column**, or **column $0$**.

Thus the entry in row $n$ and column $m$ contains the binomial coefficient $\dbinom n m$.

### Diagonal

The $n$th **diagonal** of Pascal's triangle consists of the entries $\dbinom {n + m} m$ for $m \ge 0$:

- $\dbinom n 0, \dbinom {n + 1} 1, \dbinom {n + 2} 2, \dbinom {n + 3} 3, \ldots$

Hence the **diagonal** leading down and to the right from $\dbinom 0 0$, containing all $1$s, is identified as the **zeroth diagonal**, or **diagonal $0$**.

### Lesser Diagonal

The $n$th **lesser diagonal** of Pascal's triangle consists of the entries $\dbinom {n - m} m$ for $m \ge 0$, leading up and to the right from the entry in row $n$ and column $0$:

- $\dbinom n 0, \dbinom {n - 1} 1, \dbinom {n - 2} 2, \dbinom {n - 3} 3, \ldots$

## Order of Numbers

The entries in column $n$ can be referred to as **numbers of the $n$th order (of Pascal's triangle)**, or **$n$th order numbers**.

## Also presented as

**Pascal's Triangle** is often presented in a symmetrical form, in which the columns and diagonals are both presented in a diagonal form:

While this is a visually more appealing presentation, as well as being more intuitively clear, it can be argued that it is not as straightforward for investigating its properties as the canonical presentation preferred on $\mathsf{Pr} \infty \mathsf{fWiki}$.

## Graphical Presentation

### Click here for a graphical presentation of a section of Pascal's Triangle

## Also see

- Results about
**Pascal's Triangle**can be found**here**.

## Source of Name

This entry was named for Blaise Pascal.

## Historical Note

- The earliest reference to Pascal's triangle seems to date from between the $5$th and $2$nd centuries BCE by the Hindu writer Pingala.

- The earliest known detailed discussion on it was by Halayudha in his
*Mṛtasañjīvanī*from around $1000$ CE. This was a commentary on Pingala's*Chandaḥ-sūtra*, in which it was referred to as*meru-prastaara*.

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

- In India it was discussed at length by Bhaskara II Acharya 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 (c. $1050$ CE) now lost.

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

- While the binomial coefficients for small arguments appear in works of the ancient Greeks and Romans, the first actual record of Pascal's triangle 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 used by Michael Stifel, Tartaglia and Gerolamo Cardano to calculate binomial coefficients. Tartaglia in particular used it to calculate the coefficients of the expansion of the $12$th power.

- It was Pascal's $1665$ treatise
*Traité du Triangle Arithmétique*(written in $1653$) 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 $1730$.

## Sources

- 1937: Eric Temple Bell:
*Men of Mathematics*... (previous) ... (next): Chapter $\text{V}$: "Greatness and Misery of Man" - 1964: Milton Abramowitz and Irene A. Stegun:
*Handbook of Mathematical Functions*... (previous) ... (next): $3$: Elementary Analytic Methods: $3.1$ Binomial Theorem etc.: Table of Binomial Coefficients: $3.1.8$ - 1980: David M. Burton:
*Elementary Number Theory*(revised ed.) ... (previous) ... (next): Chapter $1$: Some Preliminary Considerations: $1.2$ The Binomial Theorem - 1986: David Wells:
*Curious and Interesting Numbers*... (previous) ... (next): $24$ - 1986: David Wells:
*Curious and Interesting Numbers*... (previous) ... (next): $35$ - 1989: Ephraim J. Borowski and Jonathan M. Borwein:
*Dictionary of Mathematics*... (previous) ... (next):**Pascal's triangle** - 1997: Donald E. Knuth:
*The Art of Computer Programming: Volume 1: Fundamental Algorithms*(3rd ed.) ... (previous) ... (next): $\S 1.2.6$: Binomial Coefficients: Table $1$ - 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $24$ - 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $35$ - 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next):**Pascal's triangle** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next):**Pascal's triangle** - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next):**Pascal's triangle**

- Weisstein, Eric W. "Pascal's Triangle." From
*MathWorld*--A Wolfram Web Resource. https://mathworld.wolfram.com/PascalsTriangle.html