$($b$)(25$ points$)$ Given a non$-$negative integer M$,$ you are tasked with calculating the number

of non$-$increasing sequences p$\_(1)>=$p$\_(2)>=$p$\_(3)>=$dots of integers such that $\backslash$sum$\_$i p$\_($i$)=$M and

p$\_($i$)=\backslash$sum$\_$j $1\{$p$\_($j$)>=$i$\}$ where i is a positive integer, and $1$ is an indicator function.

Design an algorithm that computes the number of sequences with a time complexity of

O$($M$^((3)/(2))).$