I have the following triple summation: \begin{equation} \sum_{m=0}^{m_0}\sum_{j=0}^m \sum_{k=0}^{2m_0-2j} a_{kjm} x^{2(j+k)} \end{equation} I think I should be able to simplify it to something like: \begin{equation} \sum_{l=0}^{2m_0} b_l x^{2l}\,. \end{equation}
My questions is how can I relate the $a_{kjm}$ coefficient with the $b_{l}$ coefficient. Any clue is highly appreciated.
\begin{align*} \sum_{m=0}^{m_0}\sum_{j=0}^m \sum_{k=0}^{2m_0-2j}a_{kjm}x^{2(j+k)} & = \sum_{j=0}^{m_0}\sum_{m=j}^{m_0} \sum_{k=0}^{2m_0-2j}a_{kjm}x^{2(j+k)} \\ & = \sum_{j=0}^{m_0}\sum_{k=0}^{2m_0-2j} \left(\sum_{m=j}^{m_0}a_{kjm}\right)x^{2(j+k)} \\ & = \sum_{l=0}^{2m_0} \left(\sum_{j=0}^{\min\{l,2m_0-l\}} \sum_{m=j}^{m_0}a_{l-j,j,m}\right)x^{2l}. \end{align*} This uses the answer to Double Summation indexes problem. ("The", because the answers now agree!)
