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%I M1670 #46 Feb 29 2024 08:53:15
%S 1,1,2,6,24,44,80,144,264,484,888,1632,3000,5516,10144,18656,34312,
%T 63108,116072,213488,392664,722220,1328368,2443248,4493832,8265444,
%U 15202520,27961792,51429752,94594060,173985600,320009408,588589064,1082584068,1991182536
%N Rook polynomials.
%C 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
%D 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.
%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="/A004306/b004306.txt">Table of n, a(n) for n = 0..400</a>
%H N. Metropolis, M. L. Stein, and P. R. Stein, <a href="http://dx.doi.org/10.1016/S0021-9800(69)80058-X">Permanents of cyclic (0,1) matrices</a>, Journal of Combinatorial Theory, Volume 7, Issue 4, December 1969, Pages 291-321.
%H Earl Glen Whitehead, Jr., <a href="http://dx.doi.org/10.1017/S1446788700012337">Four-discordant permutations</a>, J. Austral. Math. Soc. Ser. A 28 (1979), no. 3, 369-377.
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2,0,0,-1).
%F G.f.: (1 - x + 2*x^3 + 13*x^4 - 3*x^5 - 6*x^6 - 10*x^7)/(1 - 2*x + x^4).
%F 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
%t 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 *)
%o (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
%o (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
%o (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
%Y Cf. A000803. 4th column of A008305.
%Y Equals 2 * (A001644(n) + 1), n>3.
%K nonn,easy,nice
%O 0,3
%A _N. J. A. Sloane_
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