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%I #13 Jul 29 2017 01:09:20
%S 1,8,49,280,1569,8752,48833,272976,1529441,8589176,48342449,272640680,
%T 1540495553,8718956768,49423735553,280551815456,1594568513857,
%U 9073566717800,51686272315569,294711466792120,1681938025818081,9606920311565328,54915241962566849,314131983462253680
%N Number of peaks at even level in all Schroeder paths (i.e., consisting of steps U=(1,1), D=(1,-1), H=(2,0) and never going below the axis) from (0,0) to (2n+4,0).
%C Partial sums of A026002.
%H Vincenzo Librandi, <a href="/A089383/b089383.txt">Table of n, a(n) for n = 0..200</a>
%F G.f.: (1-z-q)^2/(4*z^2*(1-z)*q), where q = sqrt(1-6*z+z^2).
%F Recurrence: (n+2)*n^2*a(n) = (n+1)*(7*n^2+4*n+1)*a(n-1) - (7*n^2+10*n+4)*n * a(n-2) + (n-1)*(n+1)^2*a(n-3). - _Vaclav Kotesovec_, Oct 24 2012
%F a(n) ~ sqrt(1632+1154*sqrt(2))*(3+2*sqrt(2))^n/(8*sqrt(Pi*n)). - _Vaclav Kotesovec_, Oct 24 2012
%e a(0) = 1 because the paths HH, HUD, UDH, UHD, UDUD and U(UD)D from (0,0) to (4,0) have only one peak at an even level (shown between parentheses).
%t CoefficientList[Series[(1-x-Sqrt[1-6*x+x^2])^2/(4*x^2*(1-x)* Sqrt[1-6*x+x^2]), {x, 0, 20}], x] (* _Vaclav Kotesovec_, Oct 24 2012 *)
%o (PARI) x='x+O('x^66); q = sqrt(1-6*x+x^2); Vec((1-x-q)^2/(4*x^2*(1-x)*q)) \\ _Joerg Arndt_, May 10 2013
%Y Cf. A006318.
%K nonn
%O 0,2
%A _Emeric Deutsch_, Dec 28 2003