A244886 G.f.: (1-x+sqrt(1-2*x-3*x^2))/(1-3*x+x^2+x^3+(1-x^2)*sqrt(1-2*x-3*x^2)).
1, 1, 2, 4, 9, 22, 56, 147, 393, 1065, 2915, 8042, 22330, 62339, 174837, 492313, 1391134, 3943130, 11207594, 31934552, 91197474, 260969372, 748176873, 2148622932, 6180146228, 17801978083, 51347929943, 148293450023, 428774359142, 1241110916678
Offset: 0
Keywords
Links
- G. C. Greubel, Table of n, a(n) for n = 0..1000 (terms 0..200 from Vincenzo Librandi)
- J.-L. Baril and A. Petrossian, Equivalence classes of Dyck paths modulo some statistics, Discrete Mathematics, Volume 338, Issue 4, 6 April 2015, Pages 655-660. See Theorem 1.
- 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.
Programs
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Mathematica
CoefficientList[Series[(1 - x + Sqrt[1 - 2 x - 3 x^2])/(1 - 3 x + x^2 + x^3 + (1 - x^2) Sqrt[1 - 2 x - 3 x^2]), {x, 0, 40}], x] (* Vincenzo Librandi, Jul 10 2014 *) -
PARI
my(x='x+O('x^50)); Vec((1-x+sqrt(1-2*x-3*x^2))/(1-3*x+x^2+x^3+(1-x^2)*sqrt(1-2*x-3*x^2))) \\ G. C. Greubel, Apr 05 2017
Formula
a(n) ~ 3^(n+7/2)/(4*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Jul 10 2014
Conjecture D-finite with recurrence: n*a(n) +(-5*n+3)*a(n-1) +2*(n)*a(n-2) +(13*n-30)*a(n-3) +3*(-1)*a(n-4) +(-8*n+21)*a(n-5) +3*(-n+3)*a(n-6)=0. - R. J. Mathar, Jan 24 2020