= Solution
Expand the normalized <Euclidean path integral> with weight $e^{-S_E}$ and apply <Wick's theorem>. Use labelled external <momenta> and a fixed <Fourier transform> convention throughout. A line carrying <momentum> $q$ contributes $(q^2+m_0^2)^{-1}$. Each $2k$-valent <interaction vertex> contributes $-\lambda_{2k}=-\mu^{d_k}g_{2k}$: its $(2k)!$ possible contractions cancel the factorial in the <Lagrangian density>. A vertex also carries $(2\pi)^D\delta^{(D)}(\sum q)$ with all incident <momenta> incoming. Integrate each independent <loop momentum> with $d^Dq/(2\pi)^D$ and divide by the <Feynman-diagram symmetry factor> for fixed external labels. There are no Minkowski factors of $i$.
For the full $N$-point <Green's function>, include every graph in which all vertices belong to components attached to external insertions. In particular, disconnected components carrying external insertions must be included, and the zeroth-order result is the sum of free pairings. Pure <vacuum bubbles> cancel against the denominator. The connected <Green's function> restricts to connected graphs, while a <one-particle-irreducible vertex> imposes the stronger restriction that removing one internal line cannot disconnect the graph. These are different objects. Odd $N$ vanishes in the symmetric <vacuum> because every interaction has even valence. At fixed perturbative order, multiply by the appropriate power of the coupling and sum all allowed graph topologies and external attachments.
If the overall momentum-conservation delta function is removed by convention, report the remaining function as its coefficient. The ordinary unamputated <Green's function> includes external <scalar propagators>; dividing them out defines an amputated function. The definition in <2/c> performs this amputation but does not, by itself, impose the usual one-particle-irreducibility condition.
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