A problem on Binomial Theorem

Question

If $C_0, C_1, C_2,..., C_n$ denote the binomial cofficients in the expansion of $(1+x)^n$ then find $$C_{0}^2+2C_{1}^2 + 3C_{2}^2 +\dots+ (n+1)C_{n}^2.$$

Answer 1

Generating function approach. We have that $$a_n:=\sum_{k=0}^n(k+1)\binom{n}{k}^2=\sum_{k=0}^n(k+1)\binom{n}{k}\cdot \binom{n}{n-k}.$$ Now note that $$\sum_{k=0}^n\binom{n}{k}x^k=(1+x)^n\quad\mbox{and}\quad \sum_{k=0}^n(k+1)\binom{n}{k}x^k=\frac{d}{dx}\left(\sum_{k=0}^n\binom{n}{k}x^{k+1}\right)=\frac{d}{dx}\left(x(1+x)^n\right).$$ Hence $$a_n=[x^n]\left((1+x)^n\cdot\frac{d}{dx}\left(x(1+x)^n\right)\right)= \left((1+x)^{2n}+nx(1+x)^{2n-1}\right).$$ Can you take it from here?

Elementary approach. Note that $kC^n_k=nC^{n-1}_{k-1}$ and therefore $$\sum_{k=0}^n \left(k+1\right)\left(C^n_k\right)^2=\sum_{k=1}^n k\left(C^n_k\right)^2+\sum_{k=0}^n \left(C^n_k\right)^2=n\sum_{k=1}^n C^{n-1}_{k-1}C^n_{n-k}+\sum_{k=0}^n C^n_k C^n_{n-k}.$$ Now apply the Vandermonde's identity.

Answer 2

We need $\displaystyle\sum_{r=0}^n(r+1)\binom nr ^2$

Now $\displaystyle(r+1)\binom nr ^2=\binom nr ^2+r\cdot\binom nr ^2=\binom nr ^2+n\binom nr\binom{n-1}{r-1}$

$$(1+x)^n(x+1)^m=(1+x)^{m+n}$$

Compare the coefficients of $x^m$ to find

$$\sum_{r=0}^{\text{min}(m,n)}\binom nr\binom mr=\binom{m+n}m$$

Set $m=n, n-1$ one by one