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Search: id:A114692
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| A114692 |
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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 returns to the x-axis (0<=k<=floor(n/2)). 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. |
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| 1, 1, 1, 2, 1, 10, 1, 40, 4, 1, 160, 36, 1, 674, 220, 8, 1, 2994, 1180, 104, 1, 13872, 6056, 848, 16, 1, 66336, 30760, 5680, 272, 1, 324898, 156632, 34528, 2768, 32, 1, 1621178, 803096, 199552, 22224, 672, 1, 8210904, 4150444, 1122736, 156528, 8192, 64, 1
(list; graph; listen)
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OFFSET
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0,4
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COMMENT
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Row n contains 1+floor(n/2) terms. Row sums are the little Schroeder numbers (A001003). Sum(k*T(n,k),k=0..floor(n/2))=A114693(n-2).
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FORMULA
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G.f.=G=1/(1-z+tz-tzR), where R=[1-z-sqrt(1-6z+z^2)]/(2z) is the g.f. of the large Schroeder numbers (A006318).
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EXAMPLE
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T(4,2)=4 because we have UUD(D)UUD(D),UUD(D)UH(D),UH(D)UUD(D) and UH(D)UH(D), where U=(1,1), D=(1,-1) and H=(2,0) (the returns to the x-axis are shown between parentheses).
Triangle starts:
1;
1;
1,2;
1,10;
1,40,4;
1,160,36;
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MAPLE
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R:=(1-z-sqrt(1-6*z+z^2))/2/z: G:=1/(1-z+t*z-t*z*R): Gser:=simplify(series(G, z=0, 16)): P[0]:=1: for n from 1 to 13 do P[n]:=coeff(Gser, z^n) od: for n from 0 to 13 do seq(coeff(t*P[n], t^j), j=1..1+floor(n/2)) od; # yields sequence in triangular form
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CROSSREFS
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Cf. A001003, A114693.
Sequence in context: A055633 A105606 A132995 this_sequence A112691 A110169 A144274
Adjacent sequences: A114689 A114690 A114691 this_sequence A114693 A114694 A114695
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KEYWORD
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nonn,tabf
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AUTHOR
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Emeric Deutsch (deutsch(AT)duke.poly.edu), Dec 26 2005
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