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%I #17 Mar 29 2017 20:25:21
%S 0,1,2,5,10,21,40,79,148,287,538,1041,1964,3811,7242,14105,26974,
%T 52713,101332,198571,383326,752837,1458268,2869131,5573286,10981597,
%U 21382196,42183395,82299994,162533193,317650712,627885751,1228966140,2431126919,4764733138,9431945577
%N Sum of lengths of initial and final horizontal segments over all dispersed Dyck paths of semilength n (i.e., over all Motzkin paths of length n with no (1,0)-steps at positive heights).
%H G. C. Greubel, <a href="/A191531/b191531.txt">Table of n, a(n) for n = 0..1000</a>
%F a(n) = Sum_{k>=0} k*A191530(n,k).
%F G.f.: z*(3-2*z-sqrt(1-4*z^2))/((1-z)^2*(1-2*z+sqrt(1-4*z^2))).
%F a(n) ~ 2^(n+3/2)/sqrt(Pi*n). - _Vaclav Kotesovec_, Mar 21 2014
%F Conjecture: +n*(n-3)*a(n) -2*(n^2-2*n-2)*a(n-1) -(3*n^2-21*n+32)*a(n-2) +2*(n-2)*(4*n-13)*a(n-3) -4*(n-2)*(n-3)*a(n-4)=0. - _R. J. Mathar_, Jun 14 2016
%e a(4)=10 because in HHHH, HHUD, HUDH, UDHH, UDUD, UUDD we have 4+2+2+2+0+0=10.
%p g := z*(3-2*z-sqrt(1-4*z^2))/((1-z)^2*(1-2*z+sqrt(1-4*z^2))): gser := series(g, z = 0, 40): seq(coeff(gser, z, n), n = 0 .. 35);
%t CoefficientList[Series[x (3-2x-Sqrt[1-4x^2])/((1-x)^2 (1-2x+ Sqrt[1-4x^2])) ,{x,0,40}],x] (* _Harvey P. Dale_, Jun 19 2011 *)
%o (PARI) z='z+O('z^50); concat([0], Vec(z*(3-2*z-sqrt(1-4*z^2))/((1-z)^2*(1-2*z+sqrt(1-4*z^2))))) \\ _G. C. Greubel_, Mar 27 2017
%Y Cf. A191530.
%K nonn
%O 0,3
%A _Emeric Deutsch_, Jun 07 2011
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