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

0,2

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

Table of n, a(n) for n=0..24.

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).

Sequence in context: A173519 A151292 A179533 * A150337 A150338 A150339

Adjacent sequences: A352170 A352171 A352172 * A352174 A352175 A352176

Keyword

nonn

Author

R. J. Mathar, Mar 07 2022

Extensions

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

Status

approved