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A244885
Expansion of (1-6*x+12*x^2-8*x^3+x^4)/((1-2*x)^2*(1-3*x+x^2)).
2
1, 1, 2, 5, 14, 41, 121, 354, 1021, 2901, 8130, 22513, 61713, 167746, 452789, 1215197, 3246050, 8637641, 22912633, 60624546, 160075117, 421960101, 1110785922, 2920883425, 7673884449, 20146907266, 52863306341, 138644338349, 363489139106, 952695494201
OFFSET
0,3
LINKS
Jean-Luc Baril, Toufik Mansour, José L. Ramírez, and Mark Shattuck, Catalan words avoiding a pattern of length four, Univ. de Bourgogne (France, 2024). See p. 3.
Jean-Luc Baril and Armen Petrossian, Equivalence classes of Dyck paths modulo some statistics, Disc. Math., Vol. 338, 4, April 2015, Pages 655-660. See Theorem 2.
Jean-Luc Baril, José L. Ramírez, and Lina M. Simbaqueba, Equivalence Classes of Skew Dyck Paths Modulo some Patterns, 2021.
K. Manes, A. Sapounakis, I. Tasoulas, and P. Tsikouras, Equivalence classes of ballot paths modulo strings of length 2 and 3, arXiv:1510.01952 [math.CO], 2015.
FORMULA
G.f.: (1 - x)*(1 - 5*x + 7*x^2 - x^3)/((1 - 2*x)^2 (1 - 3*x + x^2)).
a(n) = Fibonacci(2*n+1) - (n+1)*2^(n-2) for n>0. [Bruno Berselli, Jul 10 2014]
From Colin Barker, Apr 15 2016: (Start)
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.
a(n) = 7*a(n-1)-17*a(n-2)+16*a(n-3)-4*a(n-4) for n>4.
(End)
MATHEMATICA
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 *)
LinearRecurrence[{7, -17, 16, -4}, {1, 1, 2, 5, 14}, 50] (* Harvey P. Dale, Jun 25 2022 *)
PROG
(Magma) [IsZero(n) select 1 else Fibonacci(2*n+1)-(n+1)*2^(n-2): n in [0..40]]; // Bruno Berselli, Jul 10 2014
(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
CROSSREFS
Sequence in context: A116848 A370800 A122055 * A116845 A307466 A116849
KEYWORD
nonn,easy
AUTHOR
N. J. A. Sloane, Jul 09 2014
STATUS
approved