A361698 The number of unlabeled connected 4 regular multigraphs on n nodes with 4 external legs, loops allowed.

1, 1, 2, 8, 37, 181, 1010, 6135, 40893, 295753, 2317683, 19568427, 177397551, 1719790643, 17767328745, 194954224643, 2265042428226, 27785727158182, 358952560098959, 4871697965709175, 69309502018430799, 1031550920679805502, 16030923441853969843, 259682356008358417321, 4377679648827121988375

Offset: 0

Author

R. J. Mathar, Mar 21 2023

Keywords

• nonn

Comments

Alternatively, unlabeled connected multigraphs on n+4 nodes with 4 nodes of degree 1 and n nodes of degree 4.

Links

• H. Kleinert, A. Pelster, B. Kastening, and M. Bachmann, Recursive graphical construction of Feynman diagrams and their multiplicities in Phi^4 and Phi^2*A theory, Phys. Rev. E 62 (2) (2000), 1537 eq. (4.29)

Crossrefs

Cf. A361135 (2 legs), A085549 (no legs), A129429, A361454 (not necessarily connected).

Formula

G.f.: 1 + B(x)/(1 + C(x)) - (D(x)^2 + D(x^2))/2 where B(x), C(x) and D(x) are the g.f.s of A361454, A129429 and A361135, respectively. - Andrew Howroyd, Mar 21 2023

Extensions

Terms a(7) and beyond from Andrew Howroyd, Mar 21 2023