In the textbook <https://www.springer.com/gp/book/9783034851688> (Klassische elementare Analysis, by M. Koecher) the following elegant recurrence relation is proved for $\zeta(2n)$ (on p. 157):
$$\left(n+\frac{1}{2}\right)\zeta(2n)=\sum\limits\_{m=1}^{n-1}\zeta(2m)\,\zeta(2n-2m). \tag{1}$$  

In fact (1) is equivalent to Euler's recurrence relation for Bernoulli numbers (independently found by Ramanujan)
$$(2n+1)B\_{2n}=-\sum\limits\_{m=1}^{n-1}\binom{2n}{2m}B\_{2m}\,B\_{2n-2m}. \tag{2}$$ 
Why, In contrast to (2), (1) can seldom be found in the literature (I was able to find only <https://link.springer.com/article/10.1007/s00591-007-0022-2> that mentions (1))? Are there any other references that discuss (1)?


**P.S. In addition to juan's answer.**
G.T. Williams was not the first to state the result in this form. It can be found at least in N. Nielsen, Handbuch der theorie der gammafunktion, Leipzig:
Druck und Verlag von B.G. Teubner, 1906, p. 49. I found this reference thanks to the paper "Some identities involving the Riemann zeta function. II." by R. Sitaramachandrarao and B. Davis, Indian J. Pure Appl. Math. 17(10):1175–1186, 1986. <https://www.insa.nic.in/writereaddata/UpLoadedFiles/IJPAM/20005a50_1175.pdf> This reference also has (1) and proves (among others) an interesting generalization of (1):
$$4\sum\limits\_{i+j+k=n}\zeta(2i)\zeta(2j)\zeta(2k)=(n+1)(2n+1)\zeta(2n)-6\zeta(2)\zeta(2n-2),$$ where $n\ge 3$ and the sum extends over all ordered triples $(i,j,k)$ of positive integers satisfying $i+j+k=n$.



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. 


