Zeta Power Sum

Prove that :

$\sum_{r=0}^{n} (2^{2r}-1) \zeta(2n-2r) \zeta(2r) = 0$

Notation : $\displaystyle \zeta(n) = \sum_{k=1}^{\infty} \dfrac{1}{k^n}$ denotes the Riemann Zeta function.

Bonus : Interpret it Combinatorially, Linear Algebraically (Linear Independence etc.) and as sum of roots of an Equation.

This is a part of the set Formidable Series and Integrals

#Calculus

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Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$