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A004306
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Rook polynomials.
(Formerly M1670)
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4
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1, 1, 2, 6, 24, 44, 80, 144, 264, 484, 888, 1632, 3000, 5516, 10144, 18656, 34312, 63108, 116072, 213488, 392664, 722220, 1328368, 2443248, 4493832, 8265444, 15202520, 27961792, 51429752, 94594060, 173985600, 320009408, 588589064, 1082584068, 1991182536
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OFFSET
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0,3
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COMMENTS
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a(n) is the number of perfect matchings in the circulant graph with 2*n vertices with jumps 1 and 3. - Robert Israel, Jan 24 2019
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REFERENCES
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D. H. Lehmer, Permutations with strongly restricted displacements. Combinatorial theory and its applications, II (Proc. Colloq., Balatonfured, 1969), pp. 755-770. North-Holland, Amsterdam, 1970.
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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FORMULA
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G.f.: (1 - x + 2*x^3 + 13*x^4 - 3*x^5 - 6*x^6 - 10*x^7)/(1 - 2*x + x^4).
a(n) = 2*a(n-1) - a(n-4); a(0)=1, a(1)=1, a(2)=2, a(3)=6, a(4)=24, a(5)=44, a(6)=80, a(7)=144. - Harvey P. Dale, Dec 13 2011
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MATHEMATICA
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Join[{1, 1, 2, 6}, LinearRecurrence[{2, 0, 0, -1}, {24, 44, 80, 144}, 40]] (* or *) CoefficientList[Series[(1-x+2x^3+13x^4- 3x^5- 6x^6- 10x^7)/ (1-2x+ x^4), {x, 0, 40}], x] (* Harvey P. Dale, Dec 13 2011 *)
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PROG
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(PARI) my(x='x+O('x^40)); Vec((1-x+2*x^3+13*x^4-3*x^5-6*x^6-10*x^7)/(1 -2*x+x^4)) \\ G. C. Greubel, Apr 22 2019
(Magma) R<x>:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (1-x+ 2*x^3+13*x^4-3*x^5-6*x^6-10*x^7)/(1-2*x+x^4) )); // G. C. Greubel, Apr 22 2019
(Sage) ((1-x+2*x^3+13*x^4-3*x^5-6*x^6-10*x^7)/(1-2*x+x^4)).series(x, 40).coefficients(x, sparse=False) # G. C. Greubel, Apr 22 2019
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CROSSREFS
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KEYWORD
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nonn,easy,nice
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AUTHOR
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STATUS
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approved
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