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%I #34 Feb 21 2024 08:27:57
%S 1,1,2,3,5,8,14,23,41,69,125,214,393,682,1267,2223,4171,7385,13976,
%T 24935,47544,85377,163863,295900,571216,1036471,2011130,3664548,
%U 7143068,13063637,25568085,46912433,92152906,169570215,334194418,616530391,1218694221,2253451666,4466410838
%N Number of dispersed Dyck paths of length n (i.e., Motzkin paths of length n with no (1,0) steps at positive heights) with no valleys at level 0.
%H G. C. Greubel, <a href="/A191388/b191388.txt">Table of n, a(n) for n = 0..1000</a>
%H Helmut Prodinger, <a href="https://arxiv.org/abs/2402.13026">Dispersed Dyck paths revisited</a>, arXiv:2402.13026 [math.CO], 2024.
%F a(n) = A191387(n,0).
%F G.f.: (3-sqrt(1-4*z^2))/(2-3*z+z*sqrt(1-4*z^2)).
%F a(n) ~ 2^(n+5/2) * (1+(-1)^n/49) / (sqrt(Pi)*n^(3/2)). - _Vaclav Kotesovec_, Mar 21 2014
%F a(n) = 1+Sum_{i=0..(n-1)/2}(Sum_{k=0..i}((k+1)*binomial(2*i-k,i-k)*binomial(n-2*i-1,k+1))/(i+1)). - _Vladimir Kruchinin_, Mar 27 2016
%F D-finite with recurrence -n*a(n) +3*n*a(n-1) +2*(n-6)*a(n-2) +12*(-n+3)*a(n-3) +(7*n-24)*a(n-4) +4*(n-3)*a(n-6)=0. - _R. J. Mathar_, Sep 24 2021
%e a(4)=5 because we have HHHH, HHUD, HUDH, UDHH, and UUDD, where U=(1,1), H=(1,0), and D=(1,-1) (UDUD does not qualify).
%p g := (3-sqrt(1-4*z^2))/(2-3*z+z*sqrt(1-4*z^2)): gser := series(g, z = 0, 42): seq(coeff(gser, z, n), n = 0 .. 38);
%t CoefficientList[Series[(3-Sqrt[1-4*x^2])/(2-3*x+x*Sqrt[1-4*x^2]), {x, 0, 20}], x] (* _Vaclav Kotesovec_, Mar 21 2014 *)
%o (Maxima)
%o a(n):=1+sum(sum((k+1)*binomial(2*i-k,i-k)*binomial(n-2*i-1,k+1),k,0,i)/(i+1),i,0,(n-1)/2); /* _Vladimir Kruchinin_, Mar 27 2016 */
%o (PARI) x='x+O('x^99); Vec((3-sqrt(1-4*x^2))/(2-3*x+x*sqrt(1-4*x^2))) \\ _Altug Alkan_, Mar 27 2016
%Y Cf. A191387.
%K nonn
%O 0,3
%A _Emeric Deutsch_, Jun 02 2011
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