A129429 -id:A129429 - cp's OEIS frontend

A361454 Number of 4-regular multigraphs on n unlabeled nodes with 4 external legs, loops allowed.

1, 4, 17, 78, 360, 1835, 10168, 62271, 419701, 3107800, 25108419, 219982357, 2076785950, 21011123423, 226708386212, 2598075587529, 31509529248585, 403155101535686, 5426659537490872, 76655160760249052, 1133766220709242638, 17522418780011531368, 282452568669871514771, 4740645804610572971112

Offset: 1

Andrew Howroyd, Mar 21 2023

nonn

Cf. A129429, A361698 (connected).

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

Original entry on oeis.org

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

R. J. Mathar, Mar 21 2023

nonn

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

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)

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

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

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

cp's OEIS Frontend

A361454 Number of 4-regular multigraphs on n unlabeled nodes with 4 external legs, loops allowed.

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

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