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%I #17 Sep 08 2022 08:45:25
%S 0,0,1,5,23,98,405,1644,6604,26356,104746,415155,1642493,6490622,
%T 25629581,101156936,399151400,1574818496,6213255614,24515233082,
%U 96739530062,381803092580,1507141137026,5950525214360,23498966702808
%N Number of valleys strictly above the x-axis in all hill-free Dyck paths of semilength n (a hill in a Dyck path is a peak at level 1).
%H G. C. Greubel, <a href="/A119012/b119012.txt">Table of n, a(n) for n = 1..1000</a>
%H E. Deutsch and L. Shapiro, <a href="https://doi.org/10.1016/S0012-365X(01)00121-2">A survey of the Fine numbers</a>, Discrete Math., 241 (2001), 241-265.
%F a(n) = Sum_{k=0,..,n-2} k*A119011(n,k).
%F G.f.: 2*(1-3*z-(1-z)*sqrt(1-4*z))/((1+2*z+sqrt(1-4*z))^2*sqrt(1-4*z)).
%F a(n) ~ 2^(2*n+1)/(9*sqrt(Pi*n)). - _Vaclav Kotesovec_, Mar 20 2014
%F Conjecture: 2*(n-3)*(3*n-2)*(n+2)*a(n) +(-21*n^3+50*n^2-13*n-4)*a(n-1) -2*(n-1) *(2*n-1)*(3*n+1)*a(n-2)=0. - _R. J. Mathar_, Jun 22 2016
%e a(4)=5 because in the 6 (=A000957(5)) hill-free Dyck paths of semilength 4, namely UUUUDDDD, UUUD|UDDD, UUD|UUDDD, UUD|UD|UDD, UUUDD|UDD and UUDDUUDD (U=(1,1), D=(1,-1)) we have altogether 5 valleys strictly above the x-axis (indicated by |).
%p G:=2*(1-3*z-(1-z)*sqrt(1-4*z))/(1+2*z+sqrt(1-4*z))^2/sqrt(1-4*z): Gser:=series(G,z=0,35): seq(coeff(Gser,z,n),n=1..30);
%t Rest[CoefficientList[Series[2*(1-3*x-(1-x)*Sqrt[1-4*x])/(1+2*x+ Sqrt[1-4*x])^2/Sqrt[1-4*x], {x, 0, 30}], x]] (* _Vaclav Kotesovec_, Mar 20 2014 *)
%o (PARI) x='x+O('x^30); concat([0,0], Vec(2*(1-3*x-(1-x)*sqrt(1-4*x)) /( (1+2*x+sqrt(1-4*x))^2*sqrt(1-4*x)))) \\ _G. C. Greubel_, Mar 19 2017
%o (Magma) R<x>:=PowerSeriesRing(Rationals(), 30); [0,0] cat Coefficients(R!( 2*(1-3*x-(1-x)*Sqrt(1-4*x))/((1+2*x+ Sqrt(1-4*x))^2 *Sqrt(1-4*x)) )); // _G. C. Greubel_, Apr 06 2019
%o (Sage) a=(2*(1-3*x-(1-x)*sqrt(1-4*x)) /( (1+2*x+sqrt(1-4*x))^2* sqrt(1-4*x))).series(x, 30).coefficients(x, sparse=False); a[1:] # _G. C. Greubel_, Apr 06 2019
%Y Cf. A119011.
%K nonn
%O 1,4
%A _Emeric Deutsch_, May 08 2006
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