Definition:Pascal's Triangle

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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:


PascalsTriangle.gif


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 $\mathsf{Pr} \infty \mathsf{fWiki}$.


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 B.C.E. by the Hindu writer Piṅgalá.
  • The earliest known detailed discussion on it was by Halayudha in his Mṛtasañjīvanī from around $1000$ C.E. This was a commentary on Piṅgalá's Chandaḥ-sūtra, in which it was referred to as meru-prastaara.
  • 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.


Sources