A289658 - OEIS

%I #27 Aug 28 2019 09:53:15

%S 1,3,9,27,75,186,414,840,1578,2784,4662,7476,11556,17313,25245,35955,

%T 50157,68697,92559,122889,161001,208404,266808,338154,424620,528654,

%U 652980,800634,974970,1179699,1418895,1697037,2019015,2390175,2816325,3303783,3859383,4490526,5205186,6011964,6920094

%N Number of 2-Abelian equivalence classes of words of length n over an alphabet of size 3.

%H Colin Barker, <a href="/A289658/b289658.txt">Table of n, a(n) for n = 0..1000</a>

%H Julien Cassaigne, J. Karhumäki, S. Puzynina, M. A. Whiteland, <a href="https://doi.org/10.1007/978-3-662-53132-7_7">k-Abelian Equivalence and Rationality</a>, in International Conference on Developments in Language Theory DLT 2016: Developments in Language Theory, pp. 77-88, Lecture Notes in Computer Science book series (LNCS, volume 9840).

%H Julien Cassaigne, Juhani Karhumaki, Svetlana Puzynina, Markus A. Whiteland, <a href="https://doi.org/10.3233/FI-2017-1531">k-Abelian Equivalence and Rationality</a>, Fundamenta Informaticae 154 (2017) 1-30, DOI 10.3233/FI-2017-1531.

%H Juhani Karhumäki, Markus A. Whiteland, <a href="https://doi.org/10.1007/978-3-319-98355-4_4">Regularity of k-Abelian Equivalence Classes of Fixed Cardinality</a>, Adventures Between Lower Bounds and Higher Altitudes, Lecture Notes in Computer Science, Vol. 11011, Springer, Cham, 49-62.

%H <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (5,-8,0,14,-14,0,8,-5,1).

%F See Maple code.

%F From _Colin Barker_, Jul 23 2017: (Start)

%F G.f.: (1 - 2*x + 2*x^2 + 6*x^3 - 2*x^4 - x^5 - 2*x^7 - x^8 + 2*x^9) / ((1 - x)^7*(1 + x)^2).

%F a(n) = (2*n^6 + 42*n^5 + 335*n^4 - 1140*n^3 + 5828*n^2 - 4752*n + 5760) / 1920 for n>0 and even.

%F a(n) = (2*n^6 + 42*n^5 + 335*n^4 - 1140*n^3 + 5828*n^2 - 4662*n + 5355) / 1920 for n odd.

%F (End)

%p # For n >= 1:

%p P23:=proc(n) local t1;

%p t1:=n^6/960+7*n^5/320+67*n^4/384-19*n^3/32+1457*n^2/480

%p -(1569/640+3*(-1)^n/128)*n+741/256+27*(-1)^n/256;

%p simplify(t1);

%p end;

%p [seq(P23(n),n=1..46)];

%t CoefficientList[Series[(1 - 2 x + 2 x^2 + 6 x^3 - 2 x^4 - x^5 - 2 x^7 - x^8 + 2 x^9)/((1 - x)^7*(1 + x)^2), {x, 0, 40}], x] (* _Michael De Vlieger_, Nov 29 2018 *)

%t LinearRecurrence[{5,-8,0,14,-14,0,8,-5,1},{1,3,9,27,75,186,414,840,1578,2784},50] (* _Harvey P. Dale_, Aug 28 2019 *)

%o (PARI) Vec((1 - 2*x + 2*x^2 + 6*x^3 - 2*x^4 - x^5 - 2*x^7 - x^8 + 2*x^9) / ((1 - x)^7*(1 + x)^2) + O(x^50)) \\ _Colin Barker_, Jul 23 2017

%Y Cf. A289657.

%K nonn,easy

%O 0,2

%A _N. J. A. Sloane_, Jul 22 2017