cp's OEIS Frontend

A166296 Number of Dyck paths of semilength n with no UUU's and no DDD's and having no UUDUDD's starting at level 0 (U=(1,1), D=(1,-1)).

%I A166296 #12 Jul 24 2022 14:31:34
%S A166296 1,1,2,3,6,12,26,57,128,291,670,1558,3655,8639,20554,49185,118301,
%T A166296 285840,693480,1688683,4125882,10111393,24849532,61226546,151212789,
%U A166296 374271925,928254590,2306569185,5741561804,14315544330,35748249574
%N A166296 Number of Dyck paths of semilength n with no UUU's and no DDD's and having no UUDUDD's starting at level 0 (U=(1,1), D=(1,-1)).
%H A166296 G. C. Greubel, <a href="/A166296/b166296.txt">Table of n, a(n) for n = 0..1000</a>
%F A166296 a(n) = A166295(n,0).
%F A166296 G.f.: G=2/[1-z-z^2+2*z^3+sqrt(1-2z-z^2-2z^3+z^4)].
%F A166296 a(n) ~ sqrt(360 + 161*sqrt(5)) * ((3+sqrt(5))/2)^n / (8*sqrt(Pi)*n^(3/2)). - _Vaclav Kotesovec_, Mar 20 2014
%F A166296 Equivalently, a(n) ~ 5^(1/4) * phi^(2*n + 6) / (8*sqrt(Pi)*n^(3/2)), where phi = A001622 is the golden ratio. - _Vaclav Kotesovec_, Dec 07 2021
%F A166296 D-finite with recurrence 2*(n+3)*a(n) +(-5*n-9)*a(n-1) -n*a(n-2) +12*(1)*a(n-3) +3*(n-4)*a(n-4) +3*(-n+2)*a(n-6) +(n-3)*a(n-7)=0. - _R. J. Mathar_, Jul 24 2022
%e A166296 a(3)=3 because we have UDUDUD, UDUUDD, and UUDDUD.
%p A166296 G := 2/(1-z-z^2+2*z^3+sqrt(1-2*z-z^2-2*z^3+z^4)): Gser := series(G, z = 0, 35): seq(coeff(Gser, z, n), n = 0 .. 32);
%t A166296 CoefficientList[Series[2/(1-x-x^2+2*x^3+Sqrt[1-2*x-x^2-2*x^3+x^4]), {x, 0, 20}], x] (* _Vaclav Kotesovec_, Mar 20 2014 *)
%o A166296 (PARI) x='x+O('x^50); Vec(2/(1-x-x^2+2*x^3+sqrt(1-2*x-x^2-2*x^3+x^4))) \\ _G. C. Greubel_, Mar 22 2017
%Y A166296 Cf. A166295.
%K A166296 nonn
%O A166296 0,3
%A A166296 _Emeric Deutsch_, Oct 29 2009