The Binomial Theorem

(x + y)^{n} = j = 0 \sum n (j n) x^{n - j} y^{j}

where $(k n) = \frac{n !}{k ! ( n - k )!}$ is the binomial coefficient.

Properties

k = 0 \sum n (k n) = (1 + 1)^{n} = 2^{n}

k = 0 \sum n (- 1)^{k} (k n) = (- 1 + 1)^{n} = 0

k = 0 \sum n 2^{k} (k n) = 3^{n}

(k n + 1) = (k - 1 n) + (k n)

(r m + n) = k = 0 \sum r (r - j m) (k n)

If we let $m = n$ , we have

(n 2 n) = k = 0 \sum n (k n)^{2}