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A289658 Number of 2-Abelian equivalence classes of words of length n over an alphabet of size 3. 2
1, 3, 9, 27, 75, 186, 414, 840, 1578, 2784, 4662, 7476, 11556, 17313, 25245, 35955, 50157, 68697, 92559, 122889, 161001, 208404, 266808, 338154, 424620, 528654, 652980, 800634, 974970, 1179699, 1418895, 1697037, 2019015, 2390175, 2816325, 3303783, 3859383, 4490526, 5205186, 6011964, 6920094 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
LINKS
Julien Cassaigne, J. Karhumäki, S. Puzynina, M. A. Whiteland, k-Abelian Equivalence and Rationality, 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).
Julien Cassaigne, Juhani Karhumaki, Svetlana Puzynina, Markus A. Whiteland, k-Abelian Equivalence and Rationality, Fundamenta Informaticae 154 (2017) 1-30, DOI 10.3233/FI-2017-1531.
Juhani Karhumäki, Markus A. Whiteland, Regularity of k-Abelian Equivalence Classes of Fixed Cardinality, Adventures Between Lower Bounds and Higher Altitudes, Lecture Notes in Computer Science, Vol. 11011, Springer, Cham, 49-62.
FORMULA
See Maple code.
From Colin Barker, Jul 23 2017: (Start)
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).
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.
a(n) = (2*n^6 + 42*n^5 + 335*n^4 - 1140*n^3 + 5828*n^2 - 4662*n + 5355) / 1920 for n odd.
(End)
MAPLE
# For n >= 1:
P23:=proc(n) local t1;
t1:=n^6/960+7*n^5/320+67*n^4/384-19*n^3/32+1457*n^2/480
-(1569/640+3*(-1)^n/128)*n+741/256+27*(-1)^n/256;
simplify(t1);
end;
[seq(P23(n), n=1..46)];
MATHEMATICA
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 *)
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 *)
PROG
(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
CROSSREFS
Cf. A289657.
Sequence in context: A206604 A084707 A193703 * A180238 A289693 A269684
KEYWORD
nonn,easy
AUTHOR
N. J. A. Sloane, Jul 22 2017
STATUS
approved

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Last modified February 24 02:58 EST 2024. Contains 370288 sequences. (Running on oeis4.)