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%I #20 Mar 27 2017 04:00:20
%S 0,0,0,0,1,2,8,16,47,94,244,488,1186,2372,5536,11072,25147,50294,
%T 112028,224056,491870,983740,2135440,4270880,9188406,18376812,
%U 39249768,78499536,166656772,333313544,704069248,1408138496,2961699667,5923399334,12412521388,24825042776
%N Number of peaks at height >= 2 in all dispersed Dyck paths of length n (i.e., Motzkin paths of length n with no (1,0) steps at positive heights).
%C Also number of valleys (i.e., DU's) in all dispersed Dyck paths of length n. Example: a(4)=1 because in HHHH, HHUD, HUDH, UDHH, UDUD, and UUDD we have 0+0+0+0+1+0 = 1 valley.
%C Also number of doublerises (i.e., UU's) in all dispersed Dyck paths of length n. Example: a(4)=1 because in HHHH, HHUD, HUDH, UDHH, UDUD, and UUDD we have 0+0+0+0+0+1 = 1 doublerise.
%H G. C. Greubel, <a href="/A191309/b191309.txt">Table of n, a(n) for n = 0..1000</a>
%F a(2*n+1) = 2*a(2*n).
%F a(2*n+4) = A029760(n).
%F G.f.: g = 2*z^2*(1-q)/(q*(1-2*z+q)^2), where q=sqrt(1-4*z^2).
%F a(n) ~ 2^(n-3/2)*sqrt(n)/sqrt(Pi) * (1-sqrt(2*Pi/n)). - _Vaclav Kotesovec_, Mar 20 2014
%F Conjecture: n*(n-4)*a(n) +(n^2-10*n+15)*a(n-1) +2*(-5*n^2+28*n-27)*a(n-2) -4*(n-3)*(n-8) *a(n-3) +24*(n-3)*(n-4)*a(n-4)=0. - _R. J. Mathar_, Jun 14 2016
%e a(4)=1 because in HHHH, HHUD, HUDH, UDHH, UDUD, and UUDD we have 0+0+0+0+0+1 =1 peak at height >=2.
%p q := sqrt(1-4*z^2): g := 2*z^2*(1-q)/(q*(1-2*z+q)^2): gser := series(g, z = 0, 40): seq(coeff(gser, z, n), n = 0 .. 35);
%t CoefficientList[Series[2*x^2*(1-Sqrt[1-4*x^2])/(Sqrt[1-4*x^2]*(1-2*x+ Sqrt[1-4*x^2])^2), {x, 0, 20}], x] (* _Vaclav Kotesovec_, Mar 20 2014 *)
%o (PARI) x='x+O('x^50); concat([0,0,0,0], Vec(2*x^2*(1-sqrt(1-4*x^2))/(sqrt(1-4*x^2)*(1-2*x+ sqrt(1-4*x^2))^2))) \\ _G. C. Greubel_, Mar 26 2017
%Y Cf. A029760, A191308.
%K nonn
%O 0,6
%A _Emeric Deutsch_, May 30 2011
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