# Number of terms in a multiple summation expression

After applying binomial and multinomial expansions, I get a multiple summation expression in which some summation indices depends on others. I want to find how many terms in such expression (including zero value terms if there are !!!).

E.g., $$\sum_{k=1}^{N}\sum_{i=0}^{k-1}\sum_{j=0}^{2N-k+i}\sum_{l=1}^{M}A_{k}B_{k,i}C_{k,i,j}D_{l,k}$$

If you simply want to count the number of terms there will be, replace the sum by \begin{eqnarray*} \sum_{k=1}^{N}\sum_{i=0}^{k-1}\sum_{j=0}^{2N-k+i}\sum_{l=1}^{M}1. \end{eqnarray*} So the $l$ sum will simply factor and give a value of$M$. Now \begin{eqnarray*} \sum_{j=0}^{2N-k+i}1=2N-k+i+1. \end{eqnarray*} The next sum \begin{eqnarray*} \sum_{i=0}^{k-1}2N-k+i-1=(2N-k+1)k-\frac{k(k-1)}{2}. \end{eqnarray*} And the final sum \begin{eqnarray*} \sum_{k=1}^{N}\left((2N-k-1)k-\frac{k(k-1)}{2}\right)=\frac{N(N+1)}{2}(2N+1)-\frac{N(N+1)(2N+1)}{6}-\frac{(N-1)N(N+1)}{6} \end{eqnarray*} This simplifies to $\frac{N(N+1)^2}{2}$. so there will be $\frac{MN(N+1)^2}{2}$ terms generated by this sum. Weather the terms are zero or not will depend on knowing formulae for $A_{k}$,$B_{k,i}$,$C_{k,i,j}$ & $D_{l,k}$.