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A143954 Number of peaks in the peak plateaux of all Dyck paths of semilength n.

%I A143954 #24 Mar 23 2017 04:36:49
%S A143954 0,0,1,5,19,68,243,880,3233,12021,45119,170595,648787,2479057,9509627,
%T A143954 36598497,141246127,546433952,2118424887,8227983472,32010173957,
%U A143954 124715628852,486550020967,1900433894942,7431033132717,29085434212042
%N A143954 Number of peaks in the peak plateaux of all Dyck paths of semilength n.
%C A143954 A peak plateau is a run of consecutive peaks that is preceded by an upstep and followed by a down step; a peak consists of an upstep followed by a downstep.
%H A143954 G. C. Greubel, <a href="/A143954/b143954.txt">Table of n, a(n) for n = 0..1000</a>
%F A143954 a(n) = Sum_{k=0..n-1} k*A143953(n,k).
%F A143954 G.f.: z^2*C/[(1-z)^2*sqrt(1-4z)], where C = [1-sqrt(1-4z)]/(2z) is the Catalan function.
%F A143954 a(n) ~ 2^(2*n+1)/(9*sqrt(Pi*n)). - _Vaclav Kotesovec_, Mar 20 2014
%F A143954 a(n) = Sum_{k=1..n-1} A079309(k). - _Doug Bell_, Jun 23 2015
%F A143954 Conjecture: (-n+1)*a(n) +2*(3*n-4)*a(n-1) +(-9*n+13)*a(n-2) +2*(2*n-3)*a(n-3)=0. - _R. J. Mathar_, Jun 16 2016
%e A143954 a(3)=5 because in the peak plateaux of the Dyck paths UDUDUD, UD(UUDD), (UUDD)UD, (UUDUDD) and U(UUDD)D, shown between parentheses, we have 0 + 1 + 1 + 2 + 1 = 5 peaks.
%p A143954 C:=((1-sqrt(1-4*z))*1/2)/z: G:=z^2*C/((1-z)^2*sqrt(1-4*z)): Gser:=series(G,z= 0,30): seq(coeff(Gser,z,n),n=0..25);
%t A143954 CoefficientList[Series[x^2*((1-Sqrt[1-4*x])*1/2)/x/((1-x)^2*Sqrt[1-4*x]), {x, 0, 20}], x] (* _Vaclav Kotesovec_, Mar 20 2014 *)
%o A143954 (PARI) x='x+O('x^50); concat([0,0], Vec(x*(1-sqrt(1-4*x))/(2*(1-x)^2*sqrt(1-4*x)))) \\ _G. C. Greubel_, Mar 22 2017
%Y A143954 Cf. A143952, A143953, A079309.
%K A143954 nonn
%O A143954 0,4
%A A143954 _Emeric Deutsch_, Oct 10 2008