by the fundamental theorem of finite abelian groups, the answer is the number of partitions of 100, which according to this website is equal to 190569292.
I do not think there is a partitions formula.
The # of partitions is a very complicated combinatorics problem.
There are ways to get them using recurrence relations and all that stuff. But as far as a formula that give you an answer it is does not exist. At least I never seen one. . I am scared!