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Questions about the branch of combinatorics called graph theory (not to be used for questions concerning the graph of a function). This tag can be further specialized via using it in combination with more specialized tags such as extremal-graph-theory, spectral-graph-theory, algebraic-graph-theory, topological-graph-theory, random-graphs, graph-colorings and several others.
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Number of walks on integer lattice with self-edge at zero
Consider the graph with vertices $V=\mathbb Z$ and edges
$$E=\{(n,n+1):n\in\mathbb Z\}\cup\{(0,0)\},$$
that is,
the usual integer lattice with a self-edge at zero.
For some fixed parameters $a,b,n\in …