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Search: id:A101894
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| A101894 |
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Triangle read by rows: T(n,k) is the number of Schroeder paths of length 2n and having k peaks at odd height. |
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+0
2
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| 1, 1, 1, 3, 2, 1, 10, 8, 3, 1, 36, 34, 15, 4, 1, 137, 150, 77, 24, 5, 1, 543, 678, 399, 144, 35, 6, 1, 2219, 3116, 2073, 854, 240, 48, 7, 1, 9285, 14494, 10769, 4996, 1600, 370, 63, 8, 1, 39587, 68032, 55875, 28852, 10387, 2736, 539, 80, 9, 1, 171369, 321590, 289431
(list; table; graph; listen)
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OFFSET
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0,4
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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). Row sums are the large Schroeder numbers (A006318). Column 0 yields A002212. Column 1 yields A085362.
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FORMULA
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G.f.=G=G(t, z) satisfies z(1-tz)G^2-(1-z)(1-tz)G+1-z=0.
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EXAMPLE
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T(3,2)=3 because we have H(UD)(UD), (UD)(UD)H, and (UD)H(UD), the peaks at aodd height being shown between parentheses.
Triangle begins:
1;
1,1;
3,2,1;
10,8,3,1;
36,34,15,4,1;
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MAPLE
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G := 1/2/(-z+t*z^2)*(-1+t*z+z-t*z^2+sqrt(1-2*t*z-6*z+8*t*z^2+t^2*z^2-2*t^2*z^3+5*z^2-6*t*z^3+t^2*z^4)): Gser:=simplify(series(G, z=0, 13)):P[0]:=1: for n from 1 to 11 do P[n]:=coeff(Gser, z^n) od: for n from 0 to 11 do seq(coeff(t*P[n], t^k), k=1..n+1) od;
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CROSSREFS
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Cf. A006318, A002212, A085362, A101895.
Sequence in context: A129964 A100100 A102472 this_sequence A116071 A077756 A115080
Adjacent sequences: A101891 A101892 A101893 this_sequence A101895 A101896 A101897
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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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