= Solution
Set the external source to zero in <3/d/solution> and apply <Wick's theorem>. At a single quartic <vertex>, the four position insertions have three pairings; at a single sextic <vertex>, six insertions have $5!!=15$ pairings. Thus
$$
\boxed{\frac{Z}{Z_0}=1-\frac\lambda8\int_0^\beta d\tau\,[G^{(0)}(0)]^2
-\frac\mu{48}\int_0^\beta d\tau\,[G^{(0)}(0)]^3+O(\lambda^2,\lambda\mu,\mu^2).}
$$
Thermal time-translation invariance makes this $1-\lambda\beta g_0^2/8-\mu\beta g_0^3/48+\cdots$, with $g_0$ from <3/c/solution>. If desired the absolute corrections follow by multiplying by the free <partition function> $Z_0=[2\sinh(\beta\Omega/2)]^{-1}$.
The first <Vacuum Feynman diagram> is one quartic <vertex> carrying two self-loops, with weight $-\lambda/8$ before time integration. The second is one sextic <vertex> carrying three self-loops, with weight $-\mu/48$. No disconnected subtraction is appropriate for $Z$ itself. The top row of the following figure shows these first-order terms and the conventions for <3/f/solution>.
\Image[/past-exam-of-the-theoretical-physics-major-option-course-of-the-university-of-oxford/2016-anharmonic-oscillator-vacuum-diagrams.png]
{title=Quartic and sextic oscillator vacuum graphs through second order; the displayed coefficients include factorials and vertex interchange, and each vertex is integrated over the thermal circle}
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