Feynman diagrams related to Hedin's equations and OEIS A286784?

Question

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 ;

Comments

I haven't tried to follow the logic of the paper, but for examples of self-energy diagrams see page 6 of https://arxiv.org/abs/1811.09308

The key property is that there is no interaction between different external legs of the diagram. There are virtual particles, but they are all ultimately reabsorbed by the particle that emitted them. 

I suggest writing to the author. 

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@Mitchelporter, that link doesn't work for me. Could you give me the title of the paper? I'll also try the author once I finish this week writing up the details of the refinement of the table and its context.

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@Tom Copeland 

The paper is called "The electron self-energy in QED at two loops revisited". But the link should work now. 

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Explicit diagrams have been hand drawn and  given to me by the author Luca Molinari. I'll post an answer to this Q here when I have completed, probably this week, a set of notes on the broader context of this array.