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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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

LINKS

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

J.-L. Baril, A. Petrossian, Equivalence classes of Dyck paths modulo some statistics, Disc. Math., Vol. 338, 4, April 2015, Pages 655-660. See Theorem 2.

K. Manes, A. Sapounakis, I. Tasoulas, 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 *)

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: A116847 A116848 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

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Last modified June 22 21:29 EDT 2021. Contains 345393 sequences. (Running on oeis4.)