TITLE: How find this sum $\sum_{ab+cd=2^m}ac=?$
QUESTION [15 upvotes]: Find this sum (where $m$ is a fixed positive integer)
$$\sum_{\substack{ab+cd=2^m\\ a,b,c,d \text{ are odd}}}ac.$$
My idea: since $$ab+cd=2^m\Longrightarrow ab=2^m-cd$$
and $a,b,c,d$ is odd numbers,then
$$a=2a_{1}+1,b=2b_{1}+1,c=2c_{1}+1,d=2d_{1}+1$$
then I can't works,Thank you very  much

REPLY [2 votes]: This is mainly a comment related to the answers of Han de Bruijn and Jack D'Aurizio, but it may be of interest.
Generalizing the OP's problem to ask for
$$\sum_{\substack{ab+cd=2N\\ a,b,c,d \text{ are odd}}}ac$$
and computing the cases $2N=6$, $10$, and $12$ (to complement Han de Bruijn's computations for $2N=2$, $4$, and $8$), I get the first six terms of A007331, which are coefficients of a modular form.