I'm looking for the number of solutions (not the actual solutions) to the Diophantine equation
$\sum_{i=1}^N a_i * i = M $ of the form {$a_1$,$a_2$,...$a_n$}
where all the $a_i$ are integers greater than or equal to $0$.
I'm especially interested in the limit where integers M and N are very large (N in the range of $10^{23}$ and M in the range of $10^7$ to $10^{21}$). Ideally the solution would be in the form of Number_of_Solutions[N, M] =... Since these solutions will be very large, possibly it would be simpler to define a related function, Log_Number_of_Solutions[N, M].
(More a comment)
Look up partitions, in particular, restricted partitions.
A reasonable start:
https://en.wikipedia.org/wiki/Partition_(number_theory)#Restricted_partitions