Find the general term of the recursive relation $a_{n+1}=\frac{1}{n+1}\sum\limits_{k=0}^n a_k 2^{2n-2k+1}$ I need to find the general term of the recursive relation $a_{n+1}=\frac{1}{n+1}\sum\limits_{k=0}^n a_k 2^{2n-2k+1}$
I know it's the central binomial sequence but I can't find a way to show it.
$a_0=1$

We are given the recurrence relation
$$a_{n+1}=\frac{1}{n+1}\sum_{k=0}^{n}a_{k}2^{2n-2k+1}.$$
Then,
$$a_{n}=\frac{1}{n}\sum_{k=0}^{n-1}a_{k}2^{2n-2k-1}.$$
Multiplying the relation for $a_n$ on both sides by $\frac{2^2n}{n+1}$,
$$\frac{2^2n}{n+1}a_{n}=\frac{1}{n+1}\sum_{k=0}^{n-1}a_{k}2^{2n-2k+1}.$$
Subtracting this from the original recurrence relation, we get
$$a_{n+1}-\frac{2^2n}{n+1}a_{n} = \frac{1}{n+1}a_{n}2^{2n-2n+1} = \frac{2\,a_{n}}{n+1}\\
\implies a_{n+1} = \frac{2(2n-1)}{n+1}a_{n}.$$
Now we have a simple linear recurrence relation which may be solved by more familiar methods.
