Represent $\sum_{k=1}^{n} k^{2}$ in terms of binomial coefficient.

Question

Came across a probability problem that is sort of challenging for a beginner in a sense that I may have not seen or came across a lot of binomial identities. What I am looking for is to see if there is any way to represent $\sum_{k=1}^{n} k^{2}$ in terms of binomial coefficient.

Answer 1

Yes. Notice the following: $$k^2=k\cdot (k-1)+k=2\cdot \frac{k(k-1)}{2}+\binom{k}{1}=2\binom{k}{2}+\binom{k}{1}.$$ Doing this we have $$\sum _{k=1}^n\left (2\binom{k}{2}+\binom{k}{1}\right )=2\binom{n+1}{3}+\binom{n+1}{2},$$ using the Hockey-Stick identity.

Notice that this is part of a greater picture in which one can go from polynomials like $x^k$ to polynomials like $\binom{x}{k}$ as a change of basis.

Answer 2

The identity

$\sum_{k=0}^n k^2 = 2\binom{n+1}{3} + \binom{n+1}{2}$

given in Phicar's answer can be found by counting the following in two ways:

The number of triples of integers $(x,y,z)$ with $x < z$ and $y < z$ where $x, y$ and $z$ are among the integers $0, \dots, n$.