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A104553
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Sum of trapezoid weights of all Schroeder paths of length 2n.
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2
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1, 7, 38, 198, 1039, 5533, 29852, 162716, 893997, 4942723, 27466082, 153264066, 858230875, 4820155001, 27141345912, 153168964216, 866086326425, 4905744855359, 27830459812830, 158102366711550, 899290473825511, 5120997554408597, 29191620055374228, 166560724629655188
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OFFSET
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1,2
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COMMENTS
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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.
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LINKS
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FORMULA
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G.f.: (1-x-sqrt(1-6*x+x^2))/(2*(1-x)^2*sqrt(1-6*x+x^2)).
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
a(n) ~ sqrt(48+34*sqrt(2))*(3+2*sqrt(2))^n/(16*sqrt(Pi*n)). - Vaclav Kotesovec, Oct 24 2012
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EXAMPLE
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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.
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MAPLE
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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);
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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