Several graph sequences as solutions of a double recurrence

TL;DR

This paper explores the combinatorial structures emerging from a double recursion formula used to count connected Feynman graphs in quantum field theory, revealing interesting formulas and connections to known sequences.

Contribution

It introduces a new combinatorial approach to summing double recursions related to Feynman graph enumeration, including concise formulas and links to established sequences.

Findings

Derivation of concise formulas for the recursion sums.

Identification of a sum related to Sloane's sequence A001865.

Enhanced understanding of combinatorial structures in quantum field theory.

Abstract

We describe the combinatorics that arise in summing a double recursion formula for the enumeration of connected Feynman graphs in quantum field theory. In one index the problem is more tractable and yields concise formulas which are combinatorially interesting on their own. In the other index, one of these sums is Sloane's sequence A001865.