I just want to ask if there is a known result/proof for the given recurrence relation and multinomial expansion relationship.
The solution of the recurrence relation
$$f(m,k)=f(m-1,k)+af(m-1,k-1)+bf(m-1,k-2)+cf(m-1,k-3)$$
with initial condition $f(m,0)=1$ and $f(m,k)=0$ if $k\geq 3m+1$ is given by the $k^{th}-$term of the multinomial expansion of
$$(1+ax+bx^2+cx^3)^m.$$
I am successful in finding the solution of the recurrence of the form $f(m,k)=f(m-1,k)+af(m-1,k-1)$ given certain conditions using generating functions. However, trying to emulate the method for the problem above leads to a messy solution. So I am wondering if there are any available materials covering the problem above as a result. Or are there any possible methods to prove the problem above such that the method will provide a simple unmessy solution.
Thank you so much for your help.