  We present a systematic study of the interaction, doping, and layer
dependence of the $d_{x^2-y^2}$-wave pairing susceptibility of the Hubbard
model for a stacked 2D square lattice. We perform a multi-index perturbative
expansion up to fourth-order to obtain coefficients in powers of the Hubbard
$U$, the inter-layer $V$, and the pair-hopping $J$ interactions. We evaluate
the vertex diagrams that contribute to the pairing susceptibility for $\ell=
2,3, 4$ layered models in the U-V-J interaction space. This provides
unprecedented access to the pairing amplitudes, allowing us to identify the
processes that enhance or reduce pairing. We distinguish pairing within the
diagonal channel, $P^{\parallel}_{d}$, and off-diagonal channel,
$P^{\perp}_{d}$, and find that, in the absence of $J$, the qualitative behavior
of the layered system is equivalent to the single-layer model. In the presence
of $J$, we show that pairing is enhanced sublinearly with increasing $\ell$ and
is primarily mediated by the $P^{\perp}_{d}$ component and find which
coefficients and diagram sets are responsible. Finally, we construct a
generalized $\ell$-dependent equation for the d-wave pair susceptibility to
speculate pairing beyond $\ell=4$.
