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A114691
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Triangle read by rows: T(n,k) is the number of hill-free Schroeder paths of length 2n that have k weak ascents (1<=k<=n-1 for n>=2; k=1 for n=1). A Schroeder path of length 2n is a lattice path from (0,0) to (2n,0) consisting of U=(1,1), D=(1,-1) and H=(2,0) steps and never going below the x-axis. A hill is a peak at height 1. A weak ascent in a Schroeder path is a maximal sequence of consecutive U and H steps.
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0
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1, 3, 7, 4, 15, 26, 4, 31, 108, 54, 4, 63, 366, 380, 90, 4, 127, 1104, 1950, 960, 134, 4, 255, 3090, 8284, 6966, 2008, 186, 4, 511, 8212, 31030, 39780, 19550, 3716, 246, 4, 1023, 21014, 106252, 192802, 144472, 46670, 6308, 314, 4, 2047, 52248, 340190
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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COMMENTS
| Row n contains n-1 terms (n>=2). Row sums are the little Schroeder numbers (A001003).
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FORMULA
| G.f.=G=z(t+H)/(1-z-zH), where H is given by H =z(2+H)(t+H).
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EXAMPLE
| T(3,2)=4 because we have (UH)D(H),(UU)DD(H),(UU)D(H)D and (UU)D(U)DD, where U=(1,1), D=(1,-1) and H=(2,0) (the weak ascents are shown between parentheses).
Triangle starts:
1;
3;
7,4;
15,26,4;
31,108,54,4;
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MAPLE
| H:=(1-z*t-2*z-sqrt(1-2*z*t-4*z+z^2*t^2-4*z^2*t+4*z^2))/2/z: G:=z*(t+H)/(1-z-z*H): Gser:=simplify(series(G, z=0, 15)): for n from 1 to 11 do P[n]:=coeff(Gser, z^n) od: 1; for n from 2 to 11 do seq(coeff(P[n], t^j), j=1..n-1) od; # yields sequence in triangular form
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CROSSREFS
| Cf. A114655.
Sequence in context: A016619 A066538 A112305 * A023639 A193506 A086242
Adjacent sequences: A114688 A114689 A114690 * A114692 A114693 A114694
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KEYWORD
| nonn,tabf
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AUTHOR
| Emeric Deutsch (deutsch(AT)duke.poly.edu), Dec 24 2005
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