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Number of length n left factors of Dyck paths having no triple-rises (triple-rise = three consecutive (1,1)-steps).
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%I #9 Jun 14 2016 11:34:23

%S 1,1,2,2,4,5,9,12,22,30,55,77,141,201,368,532,974,1424,2607,3847,7043,

%T 10474,19176,28707,52559,79133,144888,219234,401420,610073,1117093,

%U 1704380,3120974,4778408,8750295,13439431,24611355,37907920,69422324,107205933,196336893

%N Number of length n left factors of Dyck paths having no triple-rises (triple-rise = three consecutive (1,1)-steps).

%C a(n)=A191785(n,0).

%F G.f.: g(z) = 2*(1+z+z^2)/(1-z^2-2*z^3+sqrt(1-2*z^2-3*z^4)).

%F a(n) ~ 3^((n+3)/2) * (11+6*sqrt(3) + (11-6*sqrt(3))*(-1)^n) / (2*n^(3/2)* sqrt(2*Pi)). - _Vaclav Kotesovec_, Mar 21 2014

%F Conjecture: -(n+3)*(13*n-70)*a(n) +(-13*n^2+19*n-102)*a(n-1) +(65*n^2-221*n-516) *a(n-2) +(65*n^2-197*n+288)*a(n-3) -(n+6)*(13*n-97) *a(n-4) +3*(-13*n^2+35*n-70) *a(n-5) +(-169*n^2+1201*n-2208) *a(n-6) -9*(13*n-40)*(n-5) *a(n-7) -6*(13*n-25)*(n-6) *a(n-8)=0. - _R. J. Mathar_, Jun 14 2016

%e a(4)=4 because we have UDUD, UDUU, UUDD, and UUDU, where U=(1,1), D=(1,-1); the paths UUUD and UUUU do not qualify.

%p g := (2*(1+z+z^2))/(1-z^2-2*z^3+sqrt(1-2*z^2-3*z^4)): gser := series(g, z = 0, 45): seq(coeff(gser, z, n), n = 0 .. 40);

%t CoefficientList[Series[(2*(1+x+x^2))/(1-x^2-2*x^3+Sqrt[1-2*x^2-3*x^4]), {x, 0, 20}], x] (* _Vaclav Kotesovec_, Mar 21 2014 *)

%Y Cf. A191785

%K nonn

%O 0,3

%A _Emeric Deutsch_, Jun 18 2011