Splitting up sums

Splitting up sums with respect to one of the allowed index numbers is not that hard to do. Let’s start by a single sum: \[\begin{equation} \sum_i^N C_i = \sum_{i \neq a}^N C_i + C_a \thinspace , \end{equation}\] and double sums are just a little more involved: \[\begin{equation} \sum_{ij}^N C_{ij} = C_{aa} + \sum_{i \neq a}^N (C_{ia} + C_{ai}) + \sum_{i \neq a, j \neq a}^N C_{ij} \thinspace . \end{equation}\] For a quadruple sum, we have \[\begin{equation} \begin{split} \sum_{ijkl}^N C_{ijkl} = &C_{aaaa} + \sum_{i \neq a}^N (C_{iaaa} + C_{aiaa} + C_{aaia} + C_{aaai}) + \sum_{i \neq a, j \neq a}^N (C_{ijaa} + C_{iaja} + C_{iaaj} + C_{aija} + C_{aiaj} + C_{aaij}) \\ &+ \sum_{i \neq a, j \neq a, k \neq a}^N (C_{ijka} + C_{ijak} + C_{iajk} + C_{aijk}) + \sum_{i \neq a, j \neq a, k \neq a, l \neq a}}^N C_{ijkl} \thinspace . \end{split} \end{equation}\]