Given the generating function $$ \sum_{n\geq 0} a_n x^n = \prod_{i=1}^n\frac{1}{1-x^{F_{i+2}-1}} $$ where $F_i$ is the $i$th Fibonacci number, how can I find what $a_n$ is?
Partial fraction decomposition seems like a possibility, but it is really messy.
Do I have to rely on asymptotics only? What can I do then?
You could Taylor expand the rhs but this will never end but you can get the $a_n$'s. DOing it, you would generate the sequence $$\{1,1,2,2,4,4,6,7,10,11,14,16,21,23,29,32,40,43,52,57,69,75,88,96,113,122,141,153, \cdots\}$$ which is $A136343$ in $OEIS$
In the links section, you will see that Alois P. Heinz provided a table of the $a_n$'s ) for $0 \leq n \leq 3500$.
Unfortunately, the $OEIS$ page does not provide much information and nothing about the asymptotics.