summation - $ sum_{r=0}^{n}binom{2n}{2r}(2-4^r)beta_{2r}=2^{2n}beta_{2n}$ - sum involving bernoulli numbers

$\displaystyle \sum_{r=0}^{n}\binom{2n}{2r}(2-4^r)\beta_{2r}=2^{2n}\beta_{2n} \\$

I am trying to prove this sum. Actually It's a conjecture made by me, And I'm pretty sure it works. Now I tried to evaluate the sum but I hardly find any way out. Any help is really appreciated .

Also the related sum $\displaystyle \sum_{r=0}^{n}\binom{2n+1}{2r}(2-4^r)\beta_{2r}=0$ is already provided but without any proof. I guess the proofs are inter related. I will make an edit if I get something

Answer

We introduce the generating function

$$G(z) = \sum_{n\ge 0} \frac{z^{2n}}{(2n)!}
\sum_{r=0}^{n} {2n\choose 2r} (2-2^{2r}) B_{2r}.$$

This is

$$\sum_{n\ge 0} z^{2n}
\sum_{r=0}^{n} \frac{1}{(2n-2r)!}
(2-2^{2r}) \frac{1}{(2r)!} B_{2r}
\\ = \sum_{n\ge 0} z^{2n}
\sum_{r=0}^{n} \frac{1}{(2n-2r)!}
(2-2^{2r}) [w^{2r}] \frac{w}{\exp(w)-1}
\\ = \sum_{r\ge 0} (2-2^{2r}) [w^{2r}] \frac{w}{\exp(w)-1}
\sum_{n\ge r} \frac{z^{2n}}{(2n-2r)!}

\\ = \sum_{r\ge 0} (2-2^{2r}) z^{2r} [w^{2r}] \frac{w}{\exp(w)-1}
\sum_{n\ge 0} \frac{z^{2n}}{(2n)!}
\\ = \frac{1}{2} (\exp(z)+\exp(-z))
\sum_{r\ge 0} (2-2^{2r}) z^{2r} [w^{2r}]
\left(\frac{1}{2} w + \frac{w}{\exp(w)-1}\right)
\\ = \frac{1}{2} (\exp(z)+\exp(-z))
\left(z + \frac{2z}{\exp(z)-1}
- z - \frac{2z}{\exp(2z)-1}
\right)
\\ = z (\exp(z)+\exp(-z))

\left(\frac{1}{\exp(z)-1}
- \frac{1}{\exp(2z)-1}
\right)
\\ = z (\exp(z)+\exp(-z))
\frac{\exp(z)}{\exp(2z)-1}
= z \frac{\exp(2z)+1}{\exp(2z)-1}
\\ = z \left(1+\frac{2}{\exp(2z)-1}\right).$$

It follows that

$$\bbox[5px,border:2px solid #00A000]{
(2n)! [z^{2n}] G(z)
= (2n)! [z^{2n}] \frac{2z}{\exp(2z)-1}}$$

which is

$$\bbox[5px,border:2px solid #00A000]{
2^{2n} (2n)! [z^{2n}] \frac{z}{\exp(z)-1}
= 2^{2n} B_{2n}}$$

and this is the claim.