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%I #19 Dec 02 2014 16:27:45
%S 1,7,38,198,1039,5533,29852,162716,893997,4942723,27466082,153264066,
%T 858230875,4820155001,27141345912,153168964216,866086326425,
%U 4905744855359,27830459812830,158102366711550,899290473825511,5120997554408597,29191620055374228,166560724629655188
%N Sum of trapezoid weights of all Schroeder paths of length 2n.
%C A Schroeder path is a lattice path starting from (0,0), ending at a point on the x-axis, consisting only of steps U=(1,1), D=(1,-1) and H=(2,0) and never going below the x-axis. Schroeder paths are counted by the large Schroeder numbers (A006318). A trapezoid in a Schroeder path is a factor of the form U^i H^j D^i (i>=1, j>=0), i being the height of the trapezoid. A trapezoid in a Schroeder path w is maximal if, as a factor in w, it is not immediately preceded by a U and immediately followed by a D. The trapezoid weight of a Schroeder path is the sum of the heights of its maximal trapezoids. For example, in the Schroeder path w=UH(UHD)D(UUDD) we have two trapezoids (shown between parentheses) of heights 1 and 2, respectively. The trapezoid weight of w is 1+2=3. This concept is an analogous to the concept of pyramid weight in a Dyck path (see the Denise-Simion paper). Partial sums of A047665 which, in turn, are the partial sums of A002002.
%H Vincenzo Librandi, <a href="/A104553/b104553.txt">Table of n, a(n) for n = 1..200</a>
%H A. Denise and R. Simion, <a href="http://dx.doi.org/10.1016/0012-365X(93)E0147-V">Two combinatorial statistics on Dyck paths</a>, Discrete Math., 137, 1995, 155-176.
%F G.f.: (1-x-sqrt(1-6*x+x^2))/(2*(1-x)^2*sqrt(1-6*x+x^2)).
%F Recurrence: n*(2*n-3)*a(n) = 2*(8*n^2 - 15*n + 5)*a(n-1) - 2*(14*n^2 - 28*n + 11)*a(n-2) + 2*(8*n^2 - 17*n + 7)*a(n-3) - (n-2)*(2*n-1)*a(n-4). - _Vaclav Kotesovec_, Oct 24 2012
%F a(n) ~ sqrt(48+34*sqrt(2))*(3+2*sqrt(2))^n/(16*sqrt(Pi*n)). - _Vaclav Kotesovec_, Oct 24 2012
%e a(2) = 7 because the six Schroeder paths of length 4, namely HH,(UD)H,H(UD),(UHD), (UD)(UD) and (UUDD), have trapezoid weights 0,1,1,1,2 and 2, respectively; the maximal trapezoids are shown between parentheses.
%p G:=(1-z-sqrt(1-6*z+z^2))/2/(1-z)^2/sqrt(1-6*z+z^2):Gser:=series(G,z=0,28): seq(coeff(Gser,z^n),n=1..25);
%t CoefficientList[Series[(1 - x - Sqrt[1 - 6 x + x^2]) / x /(2 (1 - x)^2 Sqrt[1 - 6 x + x^2]), {x, 0, 30}], x] (* _Harvey P. Dale_, May 26 2011 *)
%o (PARI) x='x+O('x^66); Vec((1-x-sqrt(1-6*x+x^2))/(2*(1-x)^2*sqrt(1-6*x+x^2))) \\ _Joerg Arndt_, May 13 2013
%Y Cf. A006318, A047665, A002002, A104552.
%K nonn
%O 1,2
%A _Emeric Deutsch_, Mar 14 2005
%E Typo in Mma program fixed by _Vincenzo Librandi_, May 13 2013
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