### Questions

For any positive integer $r$, compute $$(\frac{d}{dY})^r e^{(Y^2)}| _{Y=0}.$$ The answer should directly relates to a

**counting problem about Feynman diagrams**.Is there a tutorial for how Feynman diagrams work in this context? I look forward to an answer a lot, since the question has been reduced to the simplest form. Thank you!

**EDIT:**
3. Turns out the answer to the first question is trivial considering its Taylor expansion. So a better question should be what's the benefit of solving the first one by the combinatorial way.

### Context

I am a math student trying to learn QFT and Feynman Diagrams using Hori et al's Mirror Symmetry. Much to my surprise, even a toy model in 0-dimension the theory is already complicated (for me).

On a point, a function is just a number, so integration over all functions reduces to an ordinary integral. I am looking at the following particular toy model:

$$ \int dX e^{-S(X)}, $$

where $S(X) = \frac{1}{2}X^2 + i\epsilon X^3$, and am focusing on the perturbation with small $\epsilon$, which reduces to the computation $$(\frac{d}{dY})^r e^{(Y^2)}| _{Y=0}.$$

Basic attempts show that this is a combinatorial problem, which I have no idea how to solve. This is where the book introduces Feynman diagrams, claiming that they help computing the value at zero of the $r$-th derivative above.

However, the explanation is not clear to me. I don't know what the book means by "choosing pairs", "contracting", and "propagators". I have tried other lecture notes online, but all of what I have found use physics terminologies making the situation more complicated.

thisintegral; the point is to introduce the concept so it can be applied to more complicated integrals of the same "shape". $\endgroup$ – lambda Oct 13 at 19:51