Table 10 Convolution sums and triangular numbers

${\sum }_{m=1}^{4N-1}{\sigma }_1(m){\sigma }_1(4N-m)$

$\frac{1}{12}\{5{\sigma }_3(4N)+(1-24N){\sigma }_1(4N)\}$

${\sum }_{m=1}^{2N}{\sigma }_1(2m-1){\sigma }_1(4N-(2m-1))$

${\sigma }_3^{\ast }(2N)$

$t_8(2N-1)$

${\sum }_{\begin{array}{c}m=1\\ 4\nmid m\end{array}}^{4N-1}{\sigma }_1(m){\sigma }_1(4N-m)$

${\sigma }_3^{\ast }(2N)+{\sigma }_3^{\ast }(2N;4)$

$17t_8(N-1)$

${\sum }_{\underset{m\equiv 2mod4}{m=1\phantom{mod4}}}^{4N-1}{\sigma }_1(m){\sigma }_1(4N-m)$

${\sigma }_3^{\ast }(2N;4)$

$9t_8(N-1)$