In quantum field theory Feynman has invented a diagrammatic method to encode various terms in the Taylor decomposition of integrals of the following form below which I will write in a baby version as finite dimensional integral rather than path integral (and using "imaginary time"):
$$Z(j\_1,\dots,j\_n):=\frac{\int\_{\mathbb{R}^n} \exp\{-B(x\_1,\dots,x\_n)+\sum\_kP\_k(x)+\sum\_{i=1}^nj\_ix\_i\}dx}{ \int\_{\mathbb{R}^n} \exp\{-B(x\_1,\dots,x\_n)+\sum\_kP\_k(x)\}dx},$$
where $B$ is a positive definite quadratic form on $\mathbb{R}^n$, $P\_k$ are homogeneous polynomials.
Furthermore one can write $Z(j)=exp\{ W(j)\}$ and it is shown that $W(j)$ is a sum of terms corresponding only to **connected** diagrams.


In the context of path integrals there is a notion of effective action which in this context is defined as follows.
Let $\phi\_i:=\frac{\partial W(j)}{\partial j\_i}$. Define the Legendre transform
$$\Gamma(\phi):=\sum\_{i=1}^n \phi\_ij\_i-W(j).$$



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> In QFT it is claimed that the Taylor decomposition of $\Gamma(\phi)$ is the sum of terms corresponding to **connected one particle irreducible** diagrams. I am wondering if a finite dimensional (baby) version of this claim is true, and if this is the case whether there is a reference to a detailed discussion of the finite dimensional case.
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One particle irreducible Feynman diagrams
Section 5 of Borcherds, Barnard, [Lectures on Quantum Field Theory](https://arxiv.org/abs/math-ph/0204014) is a discussion of the 0-dimensional spacetime case, which gives finite dimensional integrals.


