Table 8 Some convolution formulas

${\sum }_{k=1}^N{\sigma }_1(2k-1){\sigma }_5(2N-(2k-1))$

$\frac{1}{408}\{768{\sigma }_7^{\ast }(N)+5b(2N)-320b(N)\}$

${\sum }_{k=1}^N{\sigma }_3(2k-1){\sigma }_3(2N-(2k-1))$

$\frac{1}{136}\{64{\sigma }_7^{\ast }(N)-b(2N)+64b(N)\}$

${\sum }_{k=1}^N{\sigma }_1(2k-1){\sigma }_7(2N-(2k-1))$

$\begin{array}{l}\frac{1}{496}\{2,176{\sigma }_9^{\ast }(N)+224d(2N)-57,344l(N)\\ +7c(2N)-1,792c(N)\}\end{array}$

${\sum }_{k=1}^N{\sigma }_3(2k-1){\sigma }_5(2N-(2k-1))$

$\begin{array}{l}\frac{1}{496}\{256{\sigma }_9^{\ast }(N)-32d(2N)+8,192l(N)\\ -c(2N)+256c(N)\}\end{array}$

${\sum }_{k=1}^N{\sigma }_5(2k-1){\sigma }_5(2N-(2k-1))$

$\begin{array}{l}\frac{1}{2,764}\{1,024{\sigma }_{11}^{\ast }(N)-\tau (2N)+1,716\tau (N)\\ -708,608\tau (\frac{N}{2})\}\end{array}$

${\sum }_{k=1}^N{\sigma }_1(2k-1){\sigma }_9(2N-(2k-1))$

$\begin{array}{l}\frac{1}{27,640}\{317,440{\sigma }_{11}^{\ast }(N)+381\tau (2N)\\ -280,656\tau (N)-112,115,712\tau (\frac{N}{2})\}\end{array}$

${\sum }_{k=1}^N{\sigma }_3(2k-1){\sigma }_7(2N-(2k-1))$

$\begin{array}{l}\frac{1}{22,112}\{17,408{\sigma }_{11}^{\ast }(N)-17\tau (2N)+4,296\tau (N)\\ +13,426,688\tau (\frac{N}{2})\}\end{array}$