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A110190
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Number of (1,0)-steps on the lines y=0 and y=1 in all Schroeder paths of length 2n (a Schroeder path of length 2n is a path from (0,0) to (2n,0), consisting of steps U=(1,1), D=(1,-1) and H=(2,0) and never going below the x-axis).
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1
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0, 1, 5, 24, 116, 568, 2820, 14184, 72180, 371112, 1925380, 10068728, 53023860, 280969560, 1497072132, 8016213960, 43114424308, 232817773640, 1261793848836, 6861179441880, 37421756333172, 204671007577464, 1122275850740996
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| a(n)=sum(k*A110189(n,k), k=0..n).
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FORMULA
| G.f.=z(1-z-2zR+z^2+2z^2*R+z^2*R^2)/(1-3z-zR+z^2+z^2*R)^2, where R=1+zR+zR^2={1-z-sqrt(1-6z+z^2)]/(2z) is the g.f. for the large Schroeder numbers (A006318).
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EXAMPLE
| a(2)=5 because in the 6 (=A006318(2)) Schroeder paths of length 4, namely, HH, HUD, UDH, UDUD, UHD, UUDD, all 5 H-steps are at levels 0 or 1.
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MAPLE
| R:=(1-z-sqrt(1-6*z+z^2))/2/z: G:=z*(1-z-2*z*R+z^2+2*z^2*R+z^2*R^2)/(1-3*z-z*R+z^2+z^2*R)^2: Gser:=series(G, z=0, 30): 0, seq(coeff(Gser, z^n), n=1..26);
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CROSSREFS
| Cf. A006318, A110189.
Sequence in context: A086347 A200739 A026707 * A026784 A017977 A017978
Adjacent sequences: A110187 A110188 A110189 * A110191 A110192 A110193
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KEYWORD
| nonn
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AUTHOR
| Emeric Deutsch (deutsch(AT)duke.poly.edu), Jul 15 2005
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