%I
%S 0,1,1,3,5,12,22,49,93,200,386,814,1586,3304,6476,13381,26333,54096,
%T 106762,218386,431910,880616,1744436,3547658,7036530,14281072,
%U 28354132,57451164,114159428,230993296,459312152,928319149,1846943453,3729244576,7423131482,14975907754
%N Number of horizontal segments in all dispersed Dyck paths of length n (i.e., in all Motzkin paths of length n with no (1,0)steps at positive heights; a horizontal segment is a maximal sequence of consecutive (1,0)steps).
%F a(n) = Sum_{k>=0} k*A191390(n,k).
%F G.f.: g(z) = 4*z*(1z)/(12*z+sqrt(14*z^2))^2.
%F a(0)=0 and a(n)=2^(n1)C(n1,floor(n/2)1) for n>=1. [_Joerg Arndt_, Aug 07 2012, aeb]
%F Dfinite with recurrence (n+1)*a(n) +(3*n1)*a(n1) +2*(n+3)*a(n2) +4*(3*n8)*a(n3) +8*(n+4)*a(n4)=0.  _R. J. Mathar_, Jul 24 2022
%e a(4)=5 because in (HHHH), (HH)UD, (H)UD(H), UD(HH), UDUD, and UUDD we have a total of 1+1+2+1+0+0=5 horizontal segments (shown between parentheses).
%p g := 4*z*(1z)/(12*z+sqrt(14*z^2))^2: gser := series(g, z = 0, 40): seq(coeff(gser, z, n), n = 0 .. 35);
%Y Cf. A191390.
%Y First differences of A045621.
%K nonn
%O 0,4
%A _Emeric Deutsch_, Jun 03 2011
