Pascal's triangle

The Pascal's triangle is a convenient tabular presentation for the binomial coefficients. Already known in the 11th century, it was adopted in Western world under this name after Blaise Pascal published his Traité du triangle arithmétique ("Treatise on the Arithmetical Triangle") in 1654.

For instance, we can use Pascal's triangle to compute the binomial expansion of

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (x+y)^4 = 1 \times a^4 + 4 \times a^3b + 6 \times a^2b^2 + 4 \times ab^3 + 1 \times b^4 ~}

The coefficients 1, 4, 6, 4, 1 appear directly in the triangle.

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{matrix} &&&&&1\\ &&&&1&&1\\ &&&1&&2&&1\\ &&1&&3&&3&&1\\ &1&&4&&6&&4&&1\\ &&&&&\cdots \end{matrix} }

Each coefficient in the triangle is the sum of the two coefficients over it. For instance, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 6 = 3 + 3} . The binomial coefficients relate to this construction by Pascal's rule, which states that if

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {n \choose k} = \frac{n!}{k! (n-k)!} }

is the kth binomial coefficient in the binomial expansion of (x+y)ⁿ, then

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {n \choose k} = {n-1 \choose k-1} + {n-1 \choose k}}

for any nonnegative integer n and any integer k between 0 and n.

Those coefficients have applications in algebra and in probabilities. From the triangle, we can equally compute the Fibonacci numbers and create the Sierpinski triangle. After studying it, Isaac Newton expanded it and found new methods to extract the square root and to calculate the natural logarithm of a number.

Properties

In order to ease the understanding of some properties, the triangle should be presented differently :

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{array}{cccccccc} 1 & & & & & & & \\ 1 & 1 & & & & & & \\ 1 & 2 & 1 & & & & & \\ 1 & 3 & 3 & 1 & & & & \\ 1 & 4 & 6 & 4 & 1 & & & \\ 1 & 5 & 10& 10& 5 & 1 & & \\ 1 & 6 & 15& 20& 15& 6 & 1 & \\ 1 & 7 & 21& 35& 35& 21& 7 & 1 \\ ...&...&...&...&...&...&...&...\\ \end{array} }

Each coefficient is the sum of the coefficient exactly over it and the other to its left. For instance, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 3 + 3 = 6} .

Let's call this rule the "addition rule".

Using this format, it is easy to apply an index to each row :

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{array}{lcccccccc} 0: & 1 & & & & & & & \\ 1: & 1 & 1 & & & & & & \\ 2: & 1 & 2 & 1 & & & & & \\ 3: & 1 & 3 & 3 & 1 & & & & \\ 4: & 1 & 4 & 6 & 4 & 1 & & & \\ 5: & 1 & 5 & 10& 10& 5 & 1 & & \\ 6: & 1 & 6 & 15& 20& 15& 6 & 1 & \\ 7: & 1 & 7 & 21& 35& 35& 21& 7 & 1 \\ ...&...&...&...&...&...&...&...&...\\ \end{array} }

Starting the indices at zero facilitates many calculations.

The sum of any row is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2^r~} , with Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r~} being the row index : Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0, 1, 2, \ldots ~} For instance, the sum of row 4 is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1 + 4 + 6 + 4 + 1 = 16 = 2^4 ~} .

Since there is a formula for summing a row, then maybe there is one for a column ? It is the case. Again, we index the triangle, but this time it will be the columns :

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{array}{lcccccccc} & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 \\ 0: & 1 & & & & & & & \\ 1: & 1 & 1 & & & & & & \\ 2: & 1 & 2 & 1 & & & & & \\ 3: & 1 & 3 & 3 & 1 & & & & \\ 4: & 1 & 4 & 6 & 4 & 1 & & & \\ 5: & 1 & 5 & 10& 10& 5 & 1 & & \\ 6: & 1 & 6 & 15& 20& 15& 6 & 1 & \\ 7: & 1 & 7 & 21& 35& 35& 21& 7 & 1 \\ ...&...&...&...&...&...&...&...&...\\ \end{array} }

Anyone familiar with the factorial function can easily find the general formula. The sum of a column Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c~} ending at row Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r~} is

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{(r+1) \times r \, \times \cdots \times (r-c+1)}{(c + 1)!} \text{ with } r, c \in \N^*} .

There is another method to compute this sum, see .

Up until now, we added along the rows and the columns. We can add along the diagonals. Adding from left to right gives :

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{array}{ccccccccccr} 1 & & & & & & & & & = & 1 \\ 1 & & & & & & & & & = & 1 \\ 1 & + & 1 & & & & & & & = & 2 \\ 1 & + & 2 & & & & & & & = & 3 \\ 1 & + & 3 & + & 1 & & & & & = & 5 \\ 1 & + & 4 & + & 3 & & & & & = & 8 \\ 1 & + & 5 & + & 6 & + & 1 & & & = & 13 \\ 1 & + & 6 & + &10 & + & 4 & + & 1 & = & 21 \\ ...& & & & & & & & & = &... \\ \end{array} }

The numbers on the rigth are the Fibonacci numbers.

One row at a time

As you kwow by now, we can build any row if we know the row before just by adding the terms two at a time. It is possible to build one row directly, without knowing the row before.

We will build the row 4 in order to discover the rule. We know that each row starts with 1 :

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{array}{lcccccc} 4: & 1 & 4 & 6 & 4 & 1 & \\ \end{array} }

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1 \times \frac{4 - 0}{1 + 0} = 4 ~}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 4 \times \frac{4 - 1}{1 + 1} = 6 ~}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 6 \times \frac{4 - 2}{1 + 2} = 4 ~}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 4 \times \frac{4 - 3}{1 + 3} = 1 ~}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1 \times \frac{4 - 4}{1 + 4} = 0 ~}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \cdots ~}