Sum of squares of combinations

Question

I am solving a PhD entrance paper at home.The problem is as follows: let $C_r=^nC_r$.Then what is the value of $C_0^2+3C_1^2+\dots+(2n+1)C_n^2$.

I have evaluated the value for $C_0^2+C_1^2+\dots+C_n^2$ which is happened to be $\frac{(2n)!}{(n!)^2}$.

Answer 1

$$ \begin{align} \sum_k(2k+1)\binom{n}{k}^2 &=\sum_k\binom{n}{k}^2+2\sum_kk\binom{n}{k}^2\tag1\\ &=\binom{2n}{n}+2n\sum_k\binom{n-1}{k-1}\binom{n}{n-k}\tag2\\ &=\binom{2n}{n}+2n\binom{2n-1}{n-1}\tag3\\[3pt] &=(n+1)\binom{2n}{n}\tag4 \end{align} $$ Explanation:
$(1)$: distribute the $2k+1$
$(2)$: $k\binom{n}{k}=n\binom{n-1}{k-1}$ and $\binom{n}{k}=\binom{n}{n-k}$ and Vandermonde's Identity
$(3)$: Vandermonde's Identity
$(4)$: $2\binom{2n-1}{n-1}=\binom{2n}{n}$

Answer 2

Generating functions:

$$ \sum_{k=0}^n (2k+1)\binom{n}{k}^2=[x^0]\left(\sum_{k=0}^n (2k+1)\binom{n}{k}x^k\right)\left(\sum_{r=0}^n \binom{n}{r} x^{-r} \right) $$

Note $[x^s]f(x)$ extracts the $x^s$ coefficient of a function $f(x)$.

The second sum you can evaluate to $(1+x^{-1})^n$, and the first is

$$ 2x\left(\sum_{k=0}^n \binom{n}{k}kx^{k-1}\right)+\sum_{k=0}^n \binom{n}{k}x^k = 2x\left[\frac{d}{dx}(1+x)^n\right]+(1+x)^n. $$

Can you proceed from here?