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A104550
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Number of horizontal segments in all Schroeder paths of length 2n (a horizontal segment is a maximal string of horizontal steps). A Schroeder path is a lattice path starting from (0,0), ending at a point on the x-axis, consisting only of steps U=(1,1), D=(1,-1) and H=(2,0) and never going below the x-axis. Schroeder paths are counted by the large Schroeder numbers (A006318).
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1
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1, 4, 20, 104, 552, 2972, 16172, 88720, 489872, 2719028, 15157188, 84799992, 475894200, 2677788492, 15102309468, 85347160608, 483183316512, 2739851422820, 15558315261812, 88462135512712, 503569008273992
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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FORMULA
| G.f.=(1-z)[1-z-sqrt(1-6z+z^2)]/[2sqrt(1-6z+z^2)]
a(n)=Jacobi_P(n+1,-1,-2,3). [From Paul Barry (pbarry(AT)wit.ie), Sep 27 2009]
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EXAMPLE
| a(2)=4 because we have (HH),(H)UD,UD(H),U(H)D,UDUD and UUDD; the 4 horizontal segments are shown between parentheses.
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MAPLE
| G:=(1-z)*(1-z-sqrt(1-6*z+z^2))/2/sqrt(1-6*z+z^2): Gser:=series(G, z=0, 28): seq(coeff(Gser, z^n), n=1..24);
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CROSSREFS
| Cf. A006318, A104549.
Cf. A035028. [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Aug 28 2008]
Sequence in context: A082761 A076035 A120978 * A035028 A089382 A192619
Adjacent sequences: A104547 A104548 A104549 * A104551 A104552 A104553
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KEYWORD
| nonn
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AUTHOR
| Emeric Deutsch (deutsch(AT)duke.poly.edu), Mar 14 2005
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