A003082 Number of multigraphs with 4 nodes and n edges.

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

Author

N. J. A. Sloane

Keywords

• easy
• nonn
• nice

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).

Crossrefs

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

Cf. A290778 (connected).

Programs

Magma

R:=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

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 *)

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

SageMath

def A003082_list(prec):
    P. = 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

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

Extensions

Entry improved by comments from Vladeta Jovovic, Dec 23 1999