# Notation for iterated summation

Is there a more compact way to write $$\sum_{i_1=0}^{N} \sum_{i_2=0}^{N-i_1} \sum_{i_3=0}^{N-i_1-i_2} \cdots \sum_{i_{K}=0}^{N-i_1-i_2-i_3-\ldots-i_{K-1}} a_{i_1i_2i_3\ldots i_K}$$ as something like $$\prod_{k=1}^K \sum_{i_{k}=0}^{N-\sum_{j=1}^{k-1}i_j} a_{i_1i_2i_3\ldots i_k}$$ for describing the iterations above instead of writing them down?

• Something like $$\sum_{i_k\ge0,i_1+\ldots+i_K\le N} a_{i_1...i_K}$$? – Dan Piponi May 17 '16 at 22:26
• It surely is more compact and describes it perfectly the operation (thanks btw), but I was expecting something more like an operator for the iteration of the sums. Do you think I should edit my question in order to make it more explicit? – Marcelo Ventura May 18 '16 at 5:14