Let $i$ and $h$ be two *adjacent* nodes in a Hamming graph and let $a$ be any positive real. Let us denote by $d_{ij}$ the distance between node $i$ and node $j$ in the graph.
I'm trying to find a compact formula for the following (finite) power series, where $j$ runs over all nodes of the graph:

$$ S= \sum_j a^{d_{ij}+d_{hj} }. \tag{1}$$

Is this a studied problem?

If the graph is an $n$-hypercube, which is the case I'm most interested in, it seems that

$$ S = \sum_{d=0}^n {{n-1}\choose {d}}a^{2d+1} + {{n-1}\choose {d-1}}a^{2d-1}\tag{2}$$

In fact, if you fix node $i$ and consider all the nodes at distance $d$ from $i$, ${{n-1}\choose {d}}$ of them are at distance $d+1$ from $h$, while the remaining ${{n-1}\choose {d-1}}$ are at distance $d-1$. However, I fail to see now how (2) can be further simplified.

I would appreciate any reference to this problem.

Thank you, M.