A recurrence relation for $\zeta(2n)$ - reference request

I have this recurrence in my collection of problems for my lessons in Analytic Number Theory. I have there the reference:

P. Ribenboim, Classical Theory of Algebraic Numbers, Springer, New York, 2001, p. 503.

I add a note that related formulas can be found in

S. Sekatskii, Novel integral representations of the Riemann zeta-function and Dirichlet eta-function, close expressions.... https://arxiv.org/abs/1606.02150 (I have not checked this paper, but contains some interesting similar relations).

After several hours I find the formula in Theorem 1 of

G. T. Williams, "A new method of evaluating $\zeta(2n)$", Amer. Math. Soc, 60, (1953) 19--25.

The author of this paper think he is the first to state the result in this form.

For a variation on the bilinear recursion relation in the OP that involves only sums up to $\lfloor n/2\rfloor$, rather than up to $n-1$, see [1,2].

Expanding the question to interesting and little known recursion relations for the zeta function I could mention this linear relation [3,4]:

$$(-1)^n\frac{\pi^{2n}n}{(2n+1)!}+\sum_{k=0}^{n-1}(-1)^k\frac{\pi^{2k}}{(2k+1)!}\zeta(2n-2k)=0.$$

[1] H.-T. Kuo, A recurrence formula for $\zeta(2n)$ (1949).
[2] L. Carlitz, A recurrence formula for $\zeta(2n)$ (1961).
[3] I. Song, A recursive formula for even order harmonic series (1987).
[4] M. Merca, On the Song recurrence relation for the Riemann zeta function (2016).