A352174 The number of connected Feynman graphs in phi^4 theory with n vertices, 2 external legs.

1, 1, 3, 10, 39, 174, 853, 4632, 27607, 180148, 1281437, 9896652, 82610706, 742066238, 7142537105, 73372890090, 801485878718, 9278500053475, 113490189831585, 1462634417672439, 19811476725948185, 281387198719498556, 4182019967162071868, 64912032433421243992, 1050391945321019422595

Offset: 0

Author

R. J. Mathar, Mar 07 2022

Keywords

• nonn

Comments

It appears (?) these are obtained from the undirected graphs of A361135 by counting graphs twice (or more often) if entering the graphs via the two external legs (marking these as an in- and an out-leg, alternatively considering their fins directed) makes a difference. There are e.g. 2 (out of 8) graphs on 3 vertices in A361135 that are not left-right-symmetric, and 9 (out of 30) on 4 vertices in A361135 which are not left-right-symmetric: a(4) = A361135(3)+2, a(5) = A361135(4)+9 (?). The index shift needed might be some sort of virtually connecting the two fins (half-edges) and considering that one more vertex. - R. J. Mathar, Mar 05 2023

a(n) is the number of connected 4-regular multigraphs on n unlabeled nodes rooted at an oriented edge, loops allowed. A361135(n) is the case for an unoriented edge. The term a(0)=1 is an artifact arising from the way the sequence was enumerated using a pair of vertices of degree 1 (see A352173). - Andrew Howroyd, Mar 10 2023

Links

• R. de Mello Koch and S. Ramgoolam, Strings from Feynman graph counting: Without large N, Phys. Rev. D 85 (2012) 026007, App. D.

Crossrefs

Cf. A352173 (not necessarily connected), A361135.

Extensions

Offset corrected and a(13) and beyond from Andrew Howroyd, Mar 10 2023