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).

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.