|
%I #28 Sep 08 2022 08:46:08
%S 1,1,2,5,14,41,121,354,1021,2901,8130,22513,61713,167746,452789,
%T 1215197,3246050,8637641,22912633,60624546,160075117,421960101,
%U 1110785922,2920883425,7673884449,20146907266,52863306341,138644338349,363489139106,952695494201
%N Expansion of (1-6*x+12*x^2-8*x^3+x^4)/((1-2*x)^2*(1-3*x+x^2)).
%H Vincenzo Librandi, <a href="/A244885/b244885.txt">Table of n, a(n) for n = 0..200</a>
%H J.-L. Baril and A. Petrossian, <a href="https://doi.org/10.1016/j.disc.2014.12.003">Equivalence classes of Dyck paths modulo some statistics</a>, Disc. Math., Vol. 338, 4, April 2015, Pages 655-660. See Theorem 2.
%H Jean-Luc Baril, José L. Ramírez, and Lina M. Simbaqueba, <a href="http://jl.baril.u-bourgogne.fr/barasi2.pdf">Equivalence Classes of Skew Dyck Paths Modulo some Patterns</a>, 2021.
%H K. Manes, A. Sapounakis, I. Tasoulas, and P. Tsikouras, <a href="http://arxiv.org/abs/1510.01952">Equivalence classes of ballot paths modulo strings of length 2 and 3</a>, arXiv:1510.01952 [math.CO], 2015.
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (7,-17,16,-4).
%F G.f.: (1 - x)*(1 - 5*x + 7*x^2 - x^3)/((1 - 2*x)^2 (1 - 3*x + x^2)).
%F a(n) = Fibonacci(2*n+1) - (n+1)*2^(n-2) for n>0. [_Bruno Berselli_, Jul 10 2014]
%F From _Colin Barker_, Apr 15 2016: (Start)
%F a(n) = ((2^(-1-n)*((3-sqrt(5))^n*(-1+sqrt(5)) + (1+sqrt(5))*(3+sqrt(5))^n))/sqrt(5) - 2^(-2+n)*(1+n)) for n>0.
%F a(n) = 7*a(n-1)-17*a(n-2)+16*a(n-3)-4*a(n-4) for n>4.
%F (End)
%t CoefficientList[Series[(1 - 6 x + 12 x^2 - 8 x^3 + x^4)/((1 - 2 x)^2 (1 - 3 x + x^2)), {x, 0, 40}], x] (* _Vincenzo Librandi_, Jul 10 2014 *)
%t LinearRecurrence[{7,-17,16,-4},{1,1,2,5,14},50] (* _Harvey P. Dale_, Jun 25 2022 *)
%o (Magma) [IsZero(n) select 1 else Fibonacci(2*n+1)-(n+1)*2^(n-2): n in [0..40]]; // _Bruno Berselli_, Jul 10 2014
%o (PARI) Vec((1-6*x+12*x^2-8*x^3+x^4)/((1-2*x)^2*(1-3*x+x^2)) + O(x^50)) \\ _Colin Barker_, Apr 15 2016
%Y Cf. A001519, A001792.
%K nonn,easy
%O 0,3
%A _N. J. A. Sloane_, Jul 09 2014
|
