A352173 The number of Feynman graphs in phi^4 theory with n vertices, 2 external legs.

1, 2, 7, 23, 85, 340, 1517, 7489, 41276, 252410, 1706071, 12660012, 102447112, 898081422, 8477941776, 85729296020, 924345402273, 10584325318278, 128259347448244, 1639694094741643, 22053783907891362, 311294619360437722, 4601020643330758040, 71063337073204684379, 1144820435086864897289

Offset: 0

Author

R. J. Mathar, Mar 07 2022

Keywords

• nonn

Comments

The generating function of this is the product of the g.f. of the connected diagrams (A352174) by the g.f. of the vacuum diagrams (A129429, including a term x^0 for the empty graph): x + 2*x^2 + 7*x^3 + 23*x^4 + ... = (x + x^2 + 3*x^3 + 10*x^4 + ...) * (1 + x + 3*x^2 + 7*x^3 + 20*x^4 + ...). - R. J. Mathar, Mar 05 2023

a(n) is the number of multigraphs with n unlabeled nodes of degree 4 plus 2 noninterchangeable nodes of degree 1, loops allowed. - 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. A352174 (connected), A129429 (0 ext. legs), A352175 (degree 3 case).

Extensions

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