# Pascal triangle

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A table of the binomial coefficients. In this table, there are 1's at the lateral sides of an equilateral triangle and each of the remaining numbers is the sum of the two numbers above it to the left and right:

$$\begin{array}{c} {} \\ {} \\ {} \\ {} \\ {} \\ {} \\ 1 \end{array} \ \begin{array}{c} {} \\ {} \\ {} \\ {} \\ {} \\ 1 \\ \cdot \end{array} \ \begin{array}{c} {} \\ {} \\ {} \\ {} \\ 1 \\ {} \\ \cdot \end{array} \ \begin{array}{c} {} \\ {} \\ {} \\ 1 \\ {} \\ 5 \\ \cdot \end{array} \ \begin{array}{c} {} \\ {} \\ 1 \\ {} \\ 4 \\ {} \\ \cdot \end{array} \ \begin{array}{c} {} \\ 1 \\ {} \\ 3 \\ {} \\ 10 \\ \cdot \end{array} \ \begin{array}{c} 1 \\ {} \\ 2 \\ {} \\ 6 \\ {} \\ \cdot \end{array} \ \begin{array}{c} {} \\ 1 \\ {} \\ 3 \\ {} \\ 10 \\ \cdot \end{array} \ \begin{array}{c} {} \\ {} \\ 1 \\ {} \\ 4 \\ {} \\ \cdot \end{array} \ \begin{array}{c} {} \\ {} \\ {} \\ 1 \\ {} \\ 5 \\ \cdot \end{array} \ \begin{array}{c} {} \\ {} \\ {} \\ {} \\ 1 \\ {} \\ \cdot \end{array} \ \begin{array}{c} {} \\ {} \\ {} \\ {} \\ {} \\ 1 \\ \cdot \end{array} \ \begin{array}{c} {} \\ {} \\ {} \\ {} \\ {} \\ {} \\ 1 \end{array}$$

In the row numbered $n+ 1$ there appear the coefficients of the expansion of the binomial $( a+ b) ^ {n}$. The triangular table presented by B. Pascal in his Treatise on an arithmetical triangle (1654) differs from the one described above by a rotation through $45\circ$. Tables for the representation of the binomial coefficients were known even earlier.

#### References

 [1] V.A. Uspenskii, "Pascal's triangle" , Moscow (1979) (In Russian) [2] , The history of mathematics from Antiquity to the beginning of the XIX-th century , 2 , Moscow (1970) (In Russian)