Chaos and Fractals, Part 2: Pascal Triangle

The notation was introduced by Albert von Ettinghausen in 1826, although these numbers were already known centuries before that (see Pascal's triangle). The function is often called the choose function, and is often read as "n choose k". The binomial coefficients are the coefficients of the series expansion of a power of a binomial. If the exponent nonnegative integer then this infinite series is actually a finite sum as all terms with k > n are 0 (zero), but if the exponent n is negative or a non-integer, then it is an infinite series. (REF: textbook 'Chaos and Fractals: the New Frontiers of Science') Applying the formula to (1 + xⁿ we immediately obtain the k^th coefficient b_k (k runs from 0 to n) of row number n of Pascal's triangle. For example, the coefficient for k = 3 in row n = 7 is: The calculation of the binomial coefficient is conveniently arranged like this: ((((5/1)·6)/2)·7)/3, alternately dividing and multiplying with increasing integers. Each division produces an integer result which is itself a binomial coefficient. The recipe to compute the coefficients of a row is thus very simple. The first and last lines are copied from the line above. These will always be equal to 1. The other coefficients are just the sum of the two coefficients in the row above. In this scheme it is most convenient to write Pascal's triangle in the form with the top vertex centered on a line above it as shown. Another identity is easy to derive: the sum of all coefficients in row number n of Pascal's triangle is equal to 2_n, which is seen by setting x = y = 1 in the binomial formula. (NOTE: while Christian Kramp (Algrebra textbook, 1808) and Euler wrote [n]; Gauss (German mathematician and physicist) used the notation π(n). (REF: Wikipedia Binomial coefficient) Pascal's rule is the important recurrence relation which follows directly from the definition: The recurrence relation just proved can be used to prove by mathematical induction that C (n, k) is a natural number (and C = combinations or choices) for all n and k. Pascal's rule also gives rise to Pascal's triangle: Row number n contains the numbers C(n,k) for k = 0,…,n. It is constructed by starting with ones at the outside and then always adding two adjacent numbers and writing the sum directly underneath. This method allows the quick calculation of binomial coefficients without the need for fractions or multiplications. For instance, by looking at row number 5 of the triangle, one can quickly read off that: (x + y)⁵ = 1 x⁵ + 5 x⁴y + 10 x³y² + 10 x²y³ + 5 x y⁴ + 1 y⁵.

The differences between elements on other diagonals are the elements in the previous diagonal, as a consequence of the recurrence relation above.

In the 1303 AD treatise Precious Mirror of the Four Elements, Zhu Shijie mentioned the triangle as an ancient method for evaluating binomial coefficients indicating that the method was known to Chinese mathematicians five centuries before Pascal.