A005816 Number of 4-valent labeled graphs with n nodes where multiple edges and loops are allowed.

1, 1, 3, 15, 138, 2021, 43581, 1295493, 50752145, 2533755933, 157055247261, 11836611005031, 1066129321651668, 113117849882149725, 13965580274228976213, 1985189312618723797371, 321932406123733248625851, 59079829666712346141491403, 12182062872168618012045410805

Offset: 0

Author

Simon Plouffe

Keywords

• nonn

Comments

Each loop contributes 2 to the valency of its node.

References

• Goulden, I. P.; Jackson, D. M.; Reilly, J. W.; The Hammond series of a symmetric function and its application to P-recursiveness. SIAM J. Algebraic Discrete Methods 4 (1983), no. 2, 179-193.
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..50 (terms 0..25 from Jason Kimberley)
• R. C. Read, The enumeration of locally restricted graphs (I), J. London Math. Soc. 34 (1959) 417-436.

Crossrefs

Column k=4 of A333467.

Cf. A005815.

Cf. A129429 (unlabeled), A033301.

Formula

a(n) = N{E_n[S_4] * S_{2n}[S_2]}.

Extensions

Definition corrected by appending "where multiple edges and loops are allowed", reference to Read 1959, formula from Read 1959 (5.11), and new terms a(16), a(17), a(18) contributed by Jason Kimberley, Jan 22 2010