  We consider spin-spin correlation functions for spins along a row, $R_n =
\langle \sigma_{0,0}\sigma_{n,0}\rangle$, in the two-dimensional Ising model.
We discuss a method for calculating general-$n$ expressions for coefficients in
high-temperature and low-temperature series expansions of $R_n$ and apply it to
obtain such expressions for several higher-order coefficients. In addition to
their intrinsic interest, these results could be useful in the continuing quest
for an ordinary differential equation whose solution would determine $R_n$,
analogous to the known ordinary differential equation whose solution determines
the diagonal correlation function $\langle \sigma_{0,0}\sigma_{n,n}\rangle$ in
this model.
