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A101275
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Triangle read by rows: T(n,k) is the number of Schroeder paths of length 2n having exactly k down steps hitting the x-axis.
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3
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1, 1, 1, 1, 4, 1, 1, 13, 7, 1, 1, 44, 34, 10, 1, 1, 165, 150, 64, 13, 1, 1, 680, 659, 346, 103, 16, 1, 1, 3001, 2973, 1753, 659, 151, 19, 1, 1, 13880, 13844, 8716, 3798, 1116, 208, 22, 1, 1, 66345, 66300, 43384, 20798, 7226, 1744, 274, 25, 1, 1, 324908, 324853, 217804
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 0,5
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COMMENTS
| 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).
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FORMULA
| G.f.=2/[2-2z-t+tz+t*sqrt(1-6z+z^2)].
1/(1-x-xy/(1-x-x/(1-x-x/(1-x-x/(1-x-x/(1-.... (continued fraction). [From Paul Barry (pbarry(AT)wit.ie), Feb 01 2009]
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EXAMPLE
| Example. T(2,1)=4 because we have UHD, UUDD, HUD and UDH.
Triangle begins:
1;
1,1;
1,4,1;
1,13,7,1;
1,44,34,10,1;
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MAPLE
| G:=2/(2-2*z-t+t*z+t*sqrt(1-6*z+z^2)): Gser:=simplify(series(G, z=0, 12)): P[0]:=1: for n from 1 to 10 do P[n]:=coeff(Gser, z^n) od: for n from 0 to 10 do seq(coeff(t*P[n], t^k), k=1..n+1) od; # yields the sequence in triangular form
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CROSSREFS
| Cf. A006318.
Sequence in context: A181145 A140070 A158815 * A039755 A047874 A080248
Adjacent sequences: A101272 A101273 A101274 * A101276 A101277 A101278
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KEYWORD
| nonn,tabl
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AUTHOR
| Emeric Deutsch (deutsch(AT)duke.poly.edu), Dec 20 2004
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