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Search: id:A101282
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| A101282 |
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Triangle read by rows: T(n,k) is the number of Schroeder paths of length 2n and having k valleys. |
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1
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| 2, 5, 1, 14, 7, 1, 42, 36, 11, 1, 132, 165, 80, 16, 1, 429, 715, 484, 155, 22, 1, 1430, 3003, 2639, 1183, 273, 29, 1, 4862, 12376, 13468, 7840, 2554, 448, 37, 1, 16796, 50388, 65688, 47328, 20124, 5031, 696, 46, 1, 58786, 203490, 310080, 267444, 141219
(list; table; graph; listen)
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OFFSET
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1,1
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COMMENT
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A Schroeder path of length 2n is a lattice path starting from (0,0), ending at (2n,0), consisting only of steps U=(1,1) (up steps), D=(1,-1) (down steps) and H=(2,0) (level steps) and never going below the x-axis (Schroeder paths are counted by the large Schroeder numbers (A006318)). Also number of Schroeder paths of length 2n and having k UU's. Also number of Schroeder paths of length 2n and having k peaks at height >1,
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FORMULA
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G.f.=G=G(t, z) satisfies z(t+z-tz)G^2-(1-2z+tz)G+1=0.
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EXAMPLE
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T(3,1)=7 because we have HU(DU)D, U(DU)DH, U(DU)HD, UH(DU)D, U(DU)UDD,
UUD(DU)D, and UU(DU)DD, the valleys being shown between parentheses.
Triangle begins:
2;
5,1;
14,7,1;
42,36,11,1;
132,165,80,16,1;
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MAPLE
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G := 1/2/(-t*z-z^2+z^2*t)*(-1+2*z-t*z+sqrt(1-4*z-2*t*z+t^2*z^2)):Gser:=simplify(series(G, z=0, 13)):for n from 1 to 11 do P[n]:=coeff(Gser, z^n) od: for n from 1 to 11 do seq(coeff(t*P[n], t^k), k=1..n) od; # yields the sequence in triangular form
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CROSSREFS
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Cf. A006318, A000108, A003516.
Sequence in context: A073187 A138159 A118919 this_sequence A145879 A101895 A110220
Adjacent sequences: A101279 A101280 A101281 this_sequence A101283 A101284 A101285
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KEYWORD
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nonn,tabl
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AUTHOR
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Emeric Deutsch (deutsch(AT)duke.poly.edu), Dec 20 2004
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