A244885 Expansion of (1-6*x+12*x^2-8*x^3+x^4)/((1-2*x)^2*(1-3*x+x^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
Links
- Vincenzo Librandi, Table of n, a(n) for n = 0..200
- 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.
- Toufik Mansour and Mark Shattuck, On ascent sequences avoiding 021 and a pattern of length four, arXiv:2507.17947 [math.CO], 2025. See p. 11.
- Index entries for linear recurrences with constant coefficients, signature (7,-17,16,-4).
Programs
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Magma
[IsZero(n) select 1 else Fibonacci(2*n+1)-(n+1)*2^(n-2): n in [0..40]]; // Bruno Berselli, Jul 10 2014
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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 *) -
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
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)