<s>[INST] Proof of $\sum_{k=0}^n{ \binom{2k}{k} 2^{2n-2k} } = (2n+1) \binom{2n}{n} = \frac{n+1}{2} \binom{2(n+1)}{n+1}$ I found some combinatorial identities in my old notebooks, but I cannot recall how I derived them. Can anyone help provide the most elementary/elegant proofs for the following identity? Specifically, the connection between the summation and either of the two closed-form expressions.
$$
\sum_{k=0}^n{ \binom{2k}{k} 2^{2n-2k} } = (2n+1) \binom{2n}{n} = \frac{n+1}{2} \binom{2(n+1)}{n+1}
$$
Some specific cases written out for some simple visualization: ($n=0$ through $n=3$)
$$
\begin{align}
\binom{0}{0} 2^{0} &= 1\binom{0}{0} = \frac{1}{2}\binom{2}{1} \\
\binom{0}{0} 2^{2} + \binom{2}{1} 2^{0} &= 3\binom{2}{1}  = \frac{2}{2}\binom{4}{2} \\
\binom{0}{0} 2^{4} + \binom{2}{1} 2^{2} + \binom{4}{2} 2^{0} &= 5\binom{4}{2} = \frac{3}{2}\binom{6}{3}\\
\binom{0}{0} 2^{6} + \binom{2}{1} 2^{4} + \binom{4}{2} 2^{2} + \binom{6}{3} 2^{0} &= 7\binom{6}{3} = \frac{4}{2}\binom{8}{4}
\end{align}
$$
EDIT: I should specify that I have already found an inductive proof, but it is not satisfying to me. I am specifically looking for a bijective/double-counting proof ideally.
 [/INST] A Couple of "Common" Series
We have the Geometric Series
$$
\sum_{k=0}^\infty 4^kx^k=\frac1{1-4x}\tag1
$$
and the Generating Function for the Central Binomial Coefficients
$$
\sum_{k=0}^\infty\binom{2k}{k}x^k=\frac1{\sqrt{1-4x}}\tag2
$$
Equation $(2)$ is the Generalized Binomial Theorem applied to $\binom{-1/2}{k}=\left(-\frac14\right)^k\binom{2k}{k}$, which is shown in this answer.

Using the Cauchy Product
We get the product of the preceding functions by taking half the derivative of $(2)$ and adjusting the index:
$$
\sum_{k=0}^\infty\frac{k+1}2\binom{2k+2}{k+1}x^k=\frac1{\sqrt{1-4x}^3}\tag3
$$
The Cauchy Product Formula then says that
$$
\begin{align}
\sum_{k=0}^n\binom{2k}{k}4^{n-k}
&=\frac{n+1}2\binom{2n+2}{n+1}\tag4\\[3pt]
&=(2n+1)\binom{2n}{n}\tag5
\end{align}
$$
</s>