I am working on some independent research and I came across a recursive sequence of the form
$$ b_{n+2} = \sum_{k=1}^{n} \binom{n}{k-1} d_{n+3-k}b_{k}, $$
with $b_{1} = 1$, $b_{2} = 0$ and $d_{j}$ has an explicit formula... Is there a way to find $b_{n}$ explicitly? Or even in general, something like
$$ a_{n+1} = \sum_{k=0}^{n-m} c_{n,k} a_{k} $$
where $a_{0}, a_{1}, \dots, a_{m} $ are given and $c_{n,k}$ is explicit.
I know generating functions but I don't believe they would work here. Thanks for the help in advance!