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%I M2543 #49 Nov 04 2022 07:31:03
%S 1,1,3,6,11,18,32,48,75,111,160,224,313,420,562,738,956,1221,1550,
%T 1936,2405,2958,3609,4368,5260,6279,7462,8814,10356,12104,14093,16320,
%U 18834,21645,24783,28272,32158,36442,41187,46410,52151,58443,65345,72864
%N Number of multigraphs with 4 nodes and n edges.
%C Also, expansion of Molien series for representation Sym^2(R^n) of the automorphism group of the lattice D_3.
%D CRC Handbook of Combinatorial Designs, 1996, p. 650.
%D J. L. Gross and J. Yellen, eds., Handbook of Graph Theory, CRC Press, 2004; p. 517.
%D F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 88, (4.1.19).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H T. D. Noe, <a href="/A003082/b003082.txt">Table of n, a(n) for n = 0..1000</a>
%H Axel Kleinschmidt and Valentin Verschinin, <a href="https://arxiv.org/abs/1706.01889">Tetrahedral modular graph functions</a>, arXiv:1706.01889 [hep-th], 2017, p. 20.
%H P. Sarnak and A. Strömbergsson, <a href="http://dx.doi.org/10.1007/s00222-005-0488-2">Minima of Epstein's zeta function and heights of flat tori</a>, Inventiones mathematicae, July 2006, Volume 165, Issue 1, pp 115-151.
%H <a href="/index/Rec#order_14">Index entries for linear recurrences with constant coefficients</a>, signature (2,0,0,-2,-2,3,0,3,-2,-2,0,0,2,-1).
%F 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).
%F 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
%F 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
%t 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 *)
%t 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 *)
%o (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
%o (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
%o (SageMath)
%o def A003082_list(prec):
%o P.<x> = PowerSeriesRing(ZZ, prec)
%o 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()
%o A003082_list(50) # _G. C. Greubel_, Nov 04 2022
%Y Cf. A001399, A014395 (5 nodes), A014396, A014397, A014398, row 4 of A192517.
%Y Cf. A290778 (connected).
%K easy,nonn,nice
%O 0,3
%A _N. J. A. Sloane_
%E Entry improved by comments from _Vladeta Jovovic_, Dec 23 1999
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