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A005822 G.f.: x*(1-x^2)*(x^4+x^3-x^2+x+1) / (x^8-4*x^6-x^4-4*x^2+1).

%I A005822 M1243 #47 Jan 05 2025 19:51:33
%S A005822 0,1,1,2,4,11,16,49,72,214,319,947,1408,4187,6223,18502,27504,81769,
%T A005822 121552,361379,537196,1597106,2374129,7058377,10492416,31194361,
%U A005822 46371025,137862866,204935836,609282227,905709904,2692710841,4002767136,11900382694,17690150767
%N A005822 G.f.: x*(1-x^2)*(x^4+x^3-x^2+x+1) / (x^8-4*x^6-x^4-4*x^2+1).
%C A005822 This is a rescaled version of the number of spanning trees in the cube of an n-cycle. See A331905 for details. - _N. J. A. Sloane_, Feb 06 2020
%D A005822 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A005822 Colin Barker, <a href="/A005822/b005822.txt">Table of n, a(n) for n = 0..1000</a> [Jul 09 2015; a(0) inserted by _Georg Fischer_, Jan 27 2020]
%H A005822 G. Baron et al., <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/23-3/baron.pdf">The number of spanning trees in the square of a cycle</a>, Fib. Quart., 23 (1985), 258-264.
%H A005822 Tsuyoshi Miezaki, <a href="/A331905/a331905.pdf">A note on spanning trees.</a>
%H A005822 Simon Plouffe, <a href="https://arxiv.org/abs/0911.4975">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
%H A005822 Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992
%H A005822 <a href="/index/Tra#trees">Index entries for sequences related to trees</a>
%H A005822 <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (0,4,0,1,0,4,0,-1).
%p A005822 A005822:=-z*(z-1)*(1+z)*(z**4+z**3-z**2+z+1)/(-4*z**6-z**4-4*z**2+1+z**8); # [Conjectured (correctly) by _Simon Plouffe_ in his 1992 dissertation; adapted to offset 0 by _Georg Fischer_, Jan 27 2020]
%t A005822 CoefficientList[Series[x (1 - x^2) (x^4 + x^3 - x^2 + x + 1) / (x^8 - 4 x^6 - x^4 - 4 x^2 + 1), {x, 0, 35}], x] (* _Vincenzo Librandi_, Jan 28 2020 *)
%o A005822 (PARI) Vec(-x*(x-1)*(x+1)*(x^4+x^3-x^2+x+1)/(x^8-4*x^6-x^4-4*x^2+1) + O(x^50)) \\ _Colin Barker_, Jul 09 2015
%o A005822 (Magma) m:=40; R<x>:=PowerSeriesRing(Rationals(), m); [0] cat Coefficients(R!( x*(1-x^2)*(x^4+x^3-x^2+x+1) / (x^8-4*x^6-x^4-4*x^2+1))); // _Vincenzo Librandi_, Jan 28 2020
%Y A005822 Cf. A169630, A331905.
%K A005822 nonn,easy
%O A005822 0,4
%A A005822 _N. J. A. Sloane_
%E A005822 G.f. adapted to the offset from _Colin Barker_, Jul 09 2015
%E A005822 Entry revised by _N. J. A. Sloane_, Jan 25 2020 and Feb 06 2020.