Evaluating the sum of $\sum_{k=0}^n \binom{n}{k}^2$

Question

I don't know if this question is trivial but let me put it in the first place. I'm trying to find the sum of $\binom{n}{0}^2+\binom{n}{1}^2+\binom{n}{2}^2+\cdots+\binom{n}{n-1}^2+\binom{n}{n}^2$ or equivalently $\displaystyle \sum_{k=0}^n \binom{n}{k}^2$.

How would you proceed?

You should not say that without providing a link, because I am pretty involved in mathematics and I did not know that. — HackerBoss, Oct 10 '18 at 15:00
Agreed. It is equal to $\binom{2n}n$, which can be proved using a combinatorial argument. — Sarvesh Ravichandran Iyer, Oct 10 '18 at 15:01
It is $$\frac{2^{2 n} \Gamma \left(n+\frac{1}{2}\right)}{\sqrt{\pi } \Gamma (n+1)}$$ — Dr. Sonnhard Graubner, Oct 10 '18 at 15:02
@HackerBoss I didn't get your statement. What link do you want? — ARahman, Oct 10 '18 at 15:08

score 1 · Answer 1 · answered Oct 10 '18 at 19:52

lets first prove a more general thing, i.e,

$\sum_{i = 0}^{x} {n \choose i } {m \choose x - i } = {m + n \choose x}$.

For that, let's suppose we have m apples and n oranges, and now we want to choose x fruits from them, we can do it in ${m + n \choose x}$ ways, and that's RHS. Now we can also select x fruits, by selecting 0 apples and x oranges, 1 apple and x - 1 oranges and so on, and that's exactly LHS, and hence LHS = RHS.

Now just put m = n, and x = n, in the equation, so it becomes

$\sum_{i = 0}^{n} {n \choose i}{n \choose n - i} = {2n \choose n}$

or,

$\sum_{i = 0}^{n} {n \choose i} {n \choose i} = {2n \choose n}$

$\sum_{i = 0}^{n} {n \choose i}^2 = {2n \choose n}$

Evaluating the sum of $\sum_{k=0}^n \binom{n}{k}^2$

1 Answers1