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A003082
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Number of multigraphs with 4 nodes and n edges.
(Formerly M2543)
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10
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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
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OFFSET
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0,3
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COMMENTS
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Also, expansion of Molien series for representation Sym^2(R^n) of the automorphism group of the lattice D_3.
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REFERENCES
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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).
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LINKS
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Index entries for linear recurrences with constant coefficients, signature (2,0,0,-2,-2,3,0,3,-2,-2,0,0,2,-1).
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FORMULA
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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
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MATHEMATICA
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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 *)
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PROG
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(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)
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()
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CROSSREFS
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KEYWORD
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easy,nonn,nice
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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