Inductive proof for $\binom{2n}{n}=\sum\limits_{k=0}^n\binom{n}{k}^2$

Solution 1:

Suppose $$ \sum_{k=0}^n\binom{n}{k}^2=\binom{2n}{n}\tag{1} $$ then $$ \begin{align} \sum_{k=0}^{n+1}\binom{n+1}{k}^2 &=\sum_{k=0}^{n+1}\left[\binom{n}{k}+\binom{n}{k-1}\right]^2\tag{2a}\\ &=\sum_{k=0}^{n+1}\left[\binom{n}{k}^2+\binom{n}{k-1}^2+2\binom{n}{k}\binom{n}{k-1}\right]\tag{2b}\\ &=\binom{2n}{n}\left(1+1+\frac{2n}{n+1}\right)\tag{2c}\\ &=\binom{2n+2}{n+1}\tag{2d} \end{align} $$ Explanation:
$\text{(2a)}$: Pascal's Triangle identity
$\text{(2b)}$: algebra
$\text{(2c)}$: apply $(1)$ and $(3)$
$\text{(2d)}$: $\binom{2n+2}{n+1}=\frac{4n+2}{n+1}\binom{2n}{n}$

Lemma: $$ \sum_{k=1}^n\binom{n}{k}\binom{n}{k-1}=\frac{n}{n+1}\sum_{k=0}^n\binom{n}{k}^2\tag{3} $$ Proof:

Since $\binom{n}{k-1}=\frac{k}{n-k+1}\binom{n}{k}$, we have $\binom{n}{k}+\binom{n}{k-1}=\frac{n+1}{n-k+1}\binom{n}{k}$. Therefore, $$ \frac{n-k+1}{n+1}\left[\binom{n}{k}+\binom{n}{k-1}\right]\binom{n}{k-1}=\binom{n}{k}\binom{n}{k-1}\tag{3a} $$ Since $\binom{n}{k}=\frac{n-k+1}{k}\binom{n}{k-1}$, we have $\binom{n}{k}+\binom{n}{k-1}=\frac{n+1}{k}\binom{n}{k-1}$. Therefore, $$ \frac{k}{n+1}\left[\binom{n}{k-1}+\binom{n}{k}\right]\binom{n}{k}=\binom{n}{k-1}\binom{n}{k}\tag{3b} $$ Adding $(3a)$ and $(3b)$ and cancelling yields $$ \frac{n-k+1}{n+1}\binom{n}{k-1}^2+\frac{k}{n+1}\binom{n}{k}^2=\binom{n}{k-1}\binom{n}{k}\tag{3c} $$ Summing $(3c)$ over $k$, and substituting $k\mapsto k+1$ in the leftmost sum, gives $$ \frac{n}{n+1}\sum_{k=0}^n\binom{n}{k}^2=\sum_{k=1}^n\binom{n}{k-1}\binom{n}{k}\tag{3d} $$ QED

Solution 2:

In general, one can show by induction that $$\frac{d^n}{dt^n} f(t)g(t) = \sum_{k=0}^n \binom{n}{k} \biggl(\frac{d^k}{dt^k}f(t) \biggr) \biggl(\frac{d^{n-k}}{dt^{n-k}}g(t) \biggr). \tag{1} $$ For $0 \le k \le r$ integers, one can see by induction that $$\frac{d^k}{dt^k} t^r = r(r-1)\cdots(r-k+1)t^{r-k} = k!\binom{r}{k}t^{r-k}. $$ So in particular, $$\frac{d^n}{dt^n} t^{2n} = n!\binom{2n}{n}t^n. \tag{2} $$ Applying $(1)$ to $f(t) = g(t) = t^n$ gives \begin{align} \frac{d^n}{dt^n} t^{2n} &= \sum_{k=0}^n \binom{n}{k} \biggl(\frac{d^k}{dt^k}t^{n} \biggr) \biggl(\frac{d^{n-k}}{dt^{n-k}}t^{n} \biggr) \\ &= \sum_{k=0}^n \binom{n}{k} \cdot k!\binom{n}{k}t^{n-k} \cdot (n-k)!\binom{n}{n-k} t^k \\ &= \sum_{k=0}^n n! \binom{n}{k}^2 t^n. \tag{3} \end{align} Comparing $(2)$ and $(3)$ gives the desired result.