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

0,3

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..200

J.-L. Baril and A. 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.

Index entries for linear recurrences with constant coefficients, signature (7,-17,16,-4).

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

Cf. A001519, A001792.

Sequence in context: A116848 A370800 A122055 * A116845 A307466 A116849

Adjacent sequences: A244882 A244883 A244884 * A244886 A244887 A244888

Keyword

nonn,easy

Author

N. J. A. Sloane, Jul 09 2014

Status

approved