OEIS A286784 [1] contains the lower triangular matrix

1;

1, 1;

2, 4, 1;

5, 15, 9, 1;

14, 56, 56, 16, 1;

... ,

with elements $T_{n,k}$ (initialized with $n=k=0$) which enumerate, according to the entry, "the number of Feynman's diagrams with $k$ fermionic loops in the order $n$ of the perturbative expansion in dimension zero for the GW approximation of the self-energy function in a many-body theory of fermions with two-body interaction." The entry references "Hedin's equations and enumeration of Feynman's diagrams" by Molinari [2] .

Does anyone have explicit graphs of the first few Feynman diagrams?

A refinement of the matrix has popped up in some algebraic combinatorics related to dual methods of compositional inversion I've explored, and I wonder if the Feynman diagrams also indicate such a refinement based on their topology.

[1]: https://oeis.org/A286784

[2]: https://arxiv.org/abs/cond-mat/0401500 ;