TITLE: Derivation of certain sums "the hard way"
QUESTION [28 upvotes]: It is a well-known fact, that one can derive some spectacular identities, e. g.

$\sum^{n-1}_{m=1}\sigma_3(m)\sigma_3(n-m)=\frac {\sigma_7(n)-\sigma_3(n)}{120}$
  $\sum^{n-1}_{m=1}\sigma_3(m)\sigma_9(n-m)=\frac {\sigma_{13}(n)-11\sigma_9(n)+10\sigma_3(n)}{2640}  
$    

just by equating several modular forms together. In the book The 1-2-3 of Modular Forms Don Zagier writes: "It is not easy to obtain any of these identities by direct number-theroretical reasoning (although in fact it can be done)"
Does anybody know how to derive these identities "the hard way" or at  least point me to some discussion?

REPLY [22 votes]: A really different proof, and the one Zagier refers to, is by Nils Skoruppa:
"A quick combinatorial proof of Eisenstein series identities", J. Number Theory 43 (1993), 68--73.