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A128748 Number of peaks at height >1 in all skew Dyck paths of semilength n.

%I A128748 #18 Jul 26 2022 11:02:48
%S A128748 0,2,11,54,260,1247,5982,28741,138364,667488,3226503,15625476,
%T A128748 75802578,368316888,1792203759,8732274312,42598366616,208036945958,
%U A128748 1017023261529,4976560342522,24372741339016,119461561111023,585970198529224
%N A128748 Number of peaks at height >1 in all skew Dyck paths of semilength n.
%C A128748 A skew Dyck path is a path in the first quadrant which begins at the origin, ends on the x-axis, consists of steps U=(1,1)(up), D=(1,-1)(down) and L=(-1,-1)(left) so that up and left steps do not overlap. The length of the path is defined to be the number of its steps.
%H A128748 G. C. Greubel, <a href="/A128748/b128748.txt">Table of n, a(n) for n = 1..1000</a>
%H A128748 E. Deutsch, E. Munarini, S. Rinaldi, <a href="http://dx.doi.org/10.1016/j.jspi.2010.01.015">Skew Dyck paths</a>, J. Stat. Plann. Infer. 140 (8) (2010) 2191-2203
%F A128748 a(n) = Sum_{k=0..n-1} A128747(n,k).
%F A128748 G.f.: (1-4*z+2*z^2+z^3-(1-z+z^2)*sqrt(1-6*z+5*z^2))/(2*z*(2-z)*sqrt(1-6*z+5*z^2)).
%F A128748 a(n) ~ 5^(n-1/2)/sqrt(Pi*n). - _Vaclav Kotesovec_, Mar 20 2014
%F A128748 D-finite with recurrence 2*(n+2)*a(n) +(-19*n-18)*a(n-1) +(53*n-12)*a(n-2) +2*(-20*n+19)*a(n-3) +(-n+26)*a(n-4) +5*(n-4)*a(n-5)=0. - _R. J. Mathar_, Jun 17 2016
%e A128748 a(2)=2 because in the paths UDUD, U(UD)D and U(UD)L we have altogether 2 peaks at height >1 (shown between parentheses).
%p A128748 G:=(1-4*z+2*z^2+z^3-(1-z+z^2)*sqrt(1-6*z+5*z^2))/2/z/(2-z)/sqrt(1-6*z+5*z^2): Gser:=series(G,z=0,30): seq(coeff(Gser,z,n),n=1..27);
%t A128748 Rest[CoefficientList[Series[(1-4*x+2*x^2+x^3-(1-x+x^2)*Sqrt[1-6*x+5*x^2]) /2/x/(2-x)/Sqrt[1-6*x+5*x^2], {x, 0, 20}], x]] (* _Vaclav Kotesovec_, Mar 20 2014 *)
%o A128748 (PARI) z='z+O('z^50); concat([0], Vec((1-4*z+2*z^2+z^3-(1-z+z^2)*sqrt(1-6*z+5*z^2))/(2*z*(2-z)*sqrt(1-6*z+5*z^2)))) \\ _G. C. Greubel_, Mar 20 2017
%Y A128748 Cf. A128747.
%K A128748 nonn
%O A128748 1,2
%A A128748 _Emeric Deutsch_, Mar 31 2007