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A191398 Number of dispersed Dyck paths of length n (i.e., Motzkin paths of length n with no (1,0)-steps at positive heights) having no DHU's (here U=(1,1), H=(1,0), and D=(1,-1)).

%I A191398 #18 Mar 27 2017 14:48:27
%S A191398 1,1,2,3,6,9,18,28,56,89,179,289,585,956,1948,3214,6591,10959,22609,
%T A191398 37833,78486,132037,275316,465255,974659,1653418,3478520,5920569,
%U A191398 12504448,21344348,45240473,77417309,164624203,282335973,602163830,1034757445,2212959172,3809387953,8167344875
%N A191398 Number of dispersed Dyck paths of length n (i.e., Motzkin paths of length n with no (1,0)-steps at positive heights) having no DHU's (here U=(1,1), H=(1,0), and D=(1,-1)).
%H A191398 G. C. Greubel, <a href="/A191398/b191398.txt">Table of n, a(n) for n = 0..1000</a>
%F A191398 a(n) = A191397(n,0).
%F A191398 G.f.: 2/(1-z-2*z^3+(1-z)*sqrt(1-4*z^2)).
%F A191398 a(n) ~ 2^(n+7/2) * (1+3*(-1)^n/49) / (sqrt(Pi)*n^(3/2)). - _Vaclav Kotesovec_, Mar 21 2014
%F A191398 Conjecture: -(n+2)*(n-3)*a(n) +(3*n^2-3*n-14)*a(n-1) +2*(n^2-7*n+8)*a(n-2) +4*(-3*n^2+12*n-10)*a(n-3) +(7*n^2-31*n+38)*a(n-4) +4*a(n-5) +4*(n-2)^2*a(n-6)=0. - _R. J. Mathar_, Jun 14 2016
%e A191398 a(5)=9 because among the 10 (=A001405(5)) dispersed Dyck paths of length 5 only UDHUD has a DHU.
%p A191398 g := 2/(1-z-2*z^3+(1-z)*sqrt(1-4*z^2)); gser := series(g, z = 0, 41); seq(coeff(gser, z, n), n = 0 .. 38);
%t A191398 CoefficientList[Series[2/(1-x-2*x^3+(1-x)*Sqrt[1-4*x^2]), {x, 0, 20}], x] (* _Vaclav Kotesovec_, Mar 21 2014 *)
%o A191398 (PARI) x='x+O('x^50); Vec(2/(1-x-2*x^3+(1-x)*sqrt(1-4*x^2))) \\ _G. C. Greubel_, Mar 26 2017
%K A191398 nonn
%O A191398 0,3
%A A191398 _Emeric Deutsch_, Jun 04 2011