This question concerns Exercise 4.24 from generatingfunctionology.
Find a three term recurrence relation, whose coefficients are polynomials in $n$, that is satisfied by $$ f(n) = (2n+11)4^n - 4(2n+1) \binom{2n}{n} $$ which is the number of convex polyominoes of perimeter $2n+8$.
It seems that the answer should be: $$ n(2n−11)f(n)=2(8n^2−42n+7)f(n−1)−8(2n−1)(2n−9)f(n−2) $$ as written in OEIS
My question is to how one would derive this recurrence relation in a systematic way? The $xD\log$ method described in the book does not seem to work immediately.