Let n be a natural number. Let dc(n) be the number of compositions of n where the summands are required to be in the set of divisors of n. Standard lore in analytic combinatorics yields the following formula for dc(n):
dc(n) = nth Taylor coefficient of 1/(1- \sum_{m ∈ divisors of n} z^m)
But what are the asymptotics of dc(n)? Here's a plot that I made:
I would like to understand the "fanning", which presumably has something to do with whether numbers have lots of small divisors or not, and I would also like to understand why these fans seem to be so close to exponentials. The solid fit line at the top is 2^n, which is the number of unrestricted compositions.