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%I #19 Feb 01 2017 10:47:33
%S 1,4,17,68,269,1056,4132,16144,63046,246228,962019,3760700,14710589,
%T 57581696,225546488,884059808,3467476430,13608852968,53443415522,
%U 210000136136,825630208466,3247733377664,12781815016232,50328168273408
%N Number of peaks at odd levels in all hill-free Dyck paths of semilength n+3 (a hill in a Dyck path is a peak at level 1).
%H Vincenzo Librandi, <a href="/A114587/b114587.txt">Table of n, a(n) for n = 0..300</a>
%F G.f.: (1 - 2*x - 3*x^2 - 2*x^3 - (1 - x^2)*sqrt(1 - 4*x))/(2*x^4*(2 + x)^2 * sqrt(1 - 4*x)).
%F a(n) = Sum_{k=0..n+1} k*A114586(n+3,k).
%F Recurrence: 8*n*(n+4)*a(n) = 2*(15*n^2 + 47*n + 18)*a(n-1) + (9*n^2 + 70*n + 80)*a(n-2) - 2*(n+1)*(2*n+1)*a(n-3). - _Vaclav Kotesovec_, Oct 19 2012
%F a(n) ~ 2^(2*n+6)/(9*sqrt(Pi*n)). - _Vaclav Kotesovec_, Oct 19 2012
%e a(1)=4 because in the 6 (=A000957(5)) hill-free Dyck paths of semilength 4, namely UUUUDDDD, UU(UD)(UD)DD, UUDU(UD)DD, UUDUDUDD, UU(UD)DUDD and UUDDUUDD (U=(1,1), D=(1,-1)) we have altogether 4 peaks at odd level (shown between parentheses).
%p G:=(1-2*z-3*z^2-2*z^3-(1-z^2)*sqrt(1-4*z))/2/sqrt(1-4*z)/z^4/(2+z)^2: Gser:=series(G,z=0,32): 1,seq(coeff(Gser,z^n),n=1..26);
%t CoefficientList[Series[(1-2*x-3*x^2-2*x^3-(1-x^2)*Sqrt[1-4*x])/(2*x^4*(2+x)^2*Sqrt[1-4*x]), {x, 0, 20}], x] (* _Vaclav Kotesovec_, Oct 19 2012 *)
%Y Cf. A114586, A114590, A114515.
%K nonn
%O 0,2
%A _Emeric Deutsch_, Dec 11 2005