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 $i+j+k=n$.