so I'm trying to come up with a procedure for maple to compute the number of planar, rooted bipartite graphs with k undirected edges using a particular recurrence relation from two generating functions.
Firstly, I have the relations for the red vertices of the graph in terms of the green vertices of the graph:
$$R_{k+1}(q):=\sum_{j=1}^{k+1} qR_{j-1}(q)G_{k+1-j}(q)$$ and $$G_{k+1}(q):=\sum_{j=1}^{k+1} G_{j-1}(q) R_{k+1-j}(q)$$ where q is a placeholder with $G_0(q)=R_0(q) = 1$.
The generating functions are for red vertices, where r = #red vertices:
$$R_k(q)= \sum_{\forall \Gamma \in D_{2k}}q^{r-1}$$ and for the $g$ green vertices in the graph.
$$ G_k(q) = \sum_{\forall \Gamma \in D_{2k}}q^g$$
$\Gamma \in D_{2k}$ is a rooted planar bipartite graph in the set of graphs in 1-1 correspondence with Dyck's paths of length $2k$ such that there are $k$ undirected edges in the graph.
My first attempt is not even worth the time to typeset to show you suffice it to say is that it is a more or less direct translation into maple's language. I've consumed the better part of the day trying to get this to work.
Suggestions are sought after and gratefully received.
Thanks.
So, abandoning the attempt to make a recursive procedure, I simply put each step into a holding vector R, and G, and with a simple for loop, I got what was needed.
R[0] := 1; G[0] := 1;
for k to 11 do
R[k] := simplify(add(G[j-1] R[k-j], j = 1 .. k));
G[k] := simplify(add(q*R[j-1]*G[k-j], j = 1 .. k))
end do
And I got precisely what I needed. I was, as per usual, using a sledgehammer, when a pair of scissors was needed.