The binomial coefficients are the coefficients produced when the sum of two unknowns is raised to successively higher powers:
(a + b)0 = 1
(a + b)1 = a + b
(a + b)2 = a2 + 2ab + b2
(a + b)3 = a3 + 3a2b + 3ab2 + b3
(a + b)4 = a4 + 4a3b + 6a2b2 + 4ab3 + b4
The coefficients on the right hand side of this sequence are
1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
Each coefficient is the sum of the two closest coefficients in the row above it. The triangle was known to Arabian scholars well before 1300 and published in China in 1303. In the European civilization Isaac Newton postulated its structure and Bernoulli found the mathematical proof in 1716.