Repeated application of simple moving average

Question

To simplify slightly, let $$S(t)=\sum_{j=-L}^L y(t+j)$$ I am looking for a single-summation formula for m repeated applications of this filter. For example, if the filter is applied twice to the sequence of y(t), \begin{eqnarray} \sum_{j=-L}^L S(t+j)&=&S(t-L)+S(t-L+1)+\cdots S(t)+S(t+1)+\cdots S(t+L) \nonumber\\ &=&y(t-2L)+2y(t-2L+1)+\cdots+(2L+1)y(t)+\cdots+2y(t+2L-1)+y(t+2L)\nonumber\\ &=&\sum_{j=-2L}^{2L} (2L+1-|j|)y(t+j) \end{eqnarray} It seems likely that this has been published - a reference would be welcome.