  Pr$_2$Ba$_4$Cu$_7$O$_{15-\delta}$ is a unique member of cuprate
superconductors where many studies suggest that CuO double chains are
responsible for superconductivity. One characteristic and non-trivial feature
of its electronic structure is a relatively large electron hopping $t$ between
nearest neighbor Cu sites with a Cu-O-Cu angle of around 90 degrees. In this
study, we have theoretically pinned down the origin of a large $|t|$ in the
double-chain structure of Pr$_2$Ba$_4$Cu$_7$O$_{15-\delta}$ using
first-principles calculation and tight-binding-model analysis. We have found
that, in the nearest neighbor hopping $t$, $d$-$d$ and $d$-$p$-$p$-$d$
contributions roughly cancel each other out and the $d$-$p$-$d$ hopping path
enhanced by the local distortion of the double chain is a key to get the large
$|t|$. Double-well band dispersion arising from the relatively large $|t/t'|$
allows the enhancement of spin-fluctuation-mediated superconductivity by the
incipient-band mechanism, where the one band bottom plays a role of the
incipient valley. Our study provides the important knowledge to understand the
unique superconductivity in Pr$_2$Ba$_4$Cu$_7$O$_{15-\delta}$.
