TITLE: How to find the asymptotics of a linear two-dimensional recurrence relation
QUESTION [8 upvotes]: Let $d$ be a positive number.
There is a two dimensional recurrence relation as follow:
$$R(n,m) = R(n-1,m-1) + R(n,m-d)$$
where
$R(0,m) = 1$ and $R(n,0) = R(n,1) = \cdots = R(n, d-1) = 1$ for all $n,m>0$.
How to analyze the asymptotics of $R(n, kn)$ for fixed $k$?
It is easy to see that
$$R(n, kn) = O\left( c_{k,d}^{n} \cdot (n+k+d)^{O(1)} \right)$$
Is there a way (or an algorithm) to find $c_{k,d}$ given $k$ and $d$?
PS: I have calcuated the bivariate generating function of $R(\cdot, \cdot)$:
\begin{align}
f(x,y) 
&= \frac{1 - xy - y^{d} + xy^{d}}{(1 - x)(1 - y)(1 - xy - y^{d})} \\
&= \frac{1}{(1 - x)(1 - y)} + \frac{xy^{d}}{(1 - x)(1 - y)(1 - xy - y^{d})} \\
\end{align}

REPLY [2 votes]: The book Analytic Combinatorics in Several Variables by Pemantle and Wilson covers problems like this extensively.