A003082 Number of multigraphs with 4 nodes and n edges. (Formerly M2543)

1, 1, 3, 6, 11, 18, 32, 48, 75, 111, 160, 224, 313, 420, 562, 738, 956, 1221, 1550, 1936, 2405, 2958, 3609, 4368, 5260, 6279, 7462, 8814, 10356, 12104, 14093, 16320, 18834, 21645, 24783, 28272, 32158, 36442, 41187, 46410, 52151, 58443, 65345, 72864

Offset

0,3

Comments

Also, expansion of Molien series for representation Sym^2(R^n) of the automorphism group of the lattice D_3.

References

CRC Handbook of Combinatorial Designs, 1996, p. 650.

J. L. Gross and J. Yellen, eds., Handbook of Graph Theory, CRC Press, 2004; p. 517.

F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 88, (4.1.19).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

T. D. Noe, Table of n, a(n) for n = 0..1000

Axel Kleinschmidt and Valentin Verschinin, Tetrahedral modular graph functions, arXiv:1706.01889 [hep-th], 2017, p. 20.

P. Sarnak and A. Strömbergsson, Minima of Epstein's zeta function and heights of flat tori, Inventiones mathematicae, July 2006, Volume 165, Issue 1, pp 115-151.

Index entries for linear recurrences with constant coefficients, signature (2,0,0,-2,-2,3,0,3,-2,-2,0,0,2,-1).

Formula

G.f.: (1-x+x^2+x^4+x^6-x^7+x^8)/((1-x)^6*(1+x)^2*(1+x^2)*(1+x+x^2)^2).

a(n) = 2*a(n-1) - 2*a(n-4) - 2*a(n-5) + 3*a(n-6) + 3*a(n-8) - 2*a(n-9) - 2*a(n-10) + 2*a(n-13) - a(n-14). - Wesley Ivan Hurt, Apr 20 2021

a(n) = (1/17280)*((3 + n)*(3175 + 2088*n + 564*n^2 + 72*n^3 + 6*n^4 + 945*(-1)^n) + 540*I^n*(1 + (-1)^n)) + (1/27)*(3*ChebyshevU(n, -1/2) + 2*ChebyshevU(n-1, -1/2) + 3*(-1)^n*(A099254(n) - A099254(n-1))). - G. C. Greubel, Nov 04 2022

Mathematica

CoefficientList[Series[PairGroupIndex[SymmetricGroup[4], s] /.Table[s[i] -> 1/(1 - x^i), {i, 1, 4}], {x, 0, 40}], x] (* Geoffrey Critzer, Nov 10 2011 *)
LinearRecurrence[{2, 0, 0, -2, -2, 3, 0, 3, -2, -2, 0, 0, 2, -1}, {1, 1, 3, 6, 11, 18, 32, 48, 75, 111, 160, 224, 313, 420}, 50] (* Harvey P. Dale, Oct 09 2016 *)

Prog

(PARI) Vec((x^8-x^7+x^6+x^4+x^2-x+1)/((x-1)^6*(x+1)^2*(x^2+1)*(x^2+x+1)^2) + O(x^100)) \\ Colin Barker, Apr 02 2015
(Magma) R<x>:=PowerSeriesRing(Integers(), 50); Coefficients(R!( (1-x+x^2+x^4+x^6-x^7+x^8)/((1-x)^6*(1+x)^2*(1+x^2)*(1+x+x^2)^2) )); // G. C. Greubel, Nov 04 2022
(SageMath)
def A003082_list(prec):
    P.<x> = PowerSeriesRing(ZZ, prec)
    return P( (1-x+x^2+x^4+x^6-x^7+x^8)/((1-x)^6*(1+x)^2*(1+x^2)*(1+x+x^2)^2) ).list()
A003082_list(50) # G. C. Greubel, Nov 04 2022

Crossrefs

Cf. A001399, A014395 (5 nodes), A014396, A014397, A014398, row 4 of A192517.

Cf. A290778 (connected).

Sequence in context: A347415 A053992 A052825 * A058053 A264923 A321381

Adjacent sequences: A003079 A003080 A003081 * A003083 A003084 A003085

Keyword

easy,nonn,nice

Author

N. J. A. Sloane

Extensions

Entry improved by comments from Vladeta Jovovic, Dec 23 1999

Status

approved