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A114589
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Number of hill-free Dyck paths of semilength n+3 and having no peaks at even levels (a hill in a Dyck path is a peak at level 1).
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1
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1, 1, 3, 7, 17, 43, 110, 286, 753, 2003, 5376, 14540, 39589, 108427, 298512, 825664, 2293271, 6393539, 17885835, 50191175, 141247519, 398537101, 1127203038, 3195229662, 9076078057, 25830193513, 73643406563, 210312889095
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| Column 0 of A114588. The number of hill-free Dyck paths having no peaks at odd level are given by the Riordan numbers (A005043).
Contribution from Paul Barry (pbarry(AT)wit.ie), Jul 05 2009: (Start)
The sequence 1,0,0,1,1,3,7,... has g.f. ((1+x)(1+2x)-sqrt((1+x)(1-3x))/(2x(2+2x+x^2)). It is
the inverse binomial transform of A035929(n+1). (End)
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FORMULA
| G.f.=[1-z-2z^2-2z^3-sqrt(1-3z^2-2z)]/[2z^4*(2+2z+z^2)].
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EXAMPLE
| a(2)=3 because we have UUUDDUUDDD, UUUDUDUDDD and UUUUUDDDDD, where U=(1,1), D=(1,-1).
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MAPLE
| G:=(1-z-2*z^2-2*z^3-sqrt(1-3*z^2-2*z))/2/z^4/(2+2*z+z^2): Gser:=series(G, z=0, 35): 1, seq(coeff(Gser, z^n), n=1..30);
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CROSSREFS
| Cf. A114588, A005043.
Sequence in context: A161943 A134184 A142975 * A192908 A078679 A025577
Adjacent sequences: A114586 A114587 A114588 * A114590 A114591 A114592
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KEYWORD
| nonn
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AUTHOR
| Emeric Deutsch (deutsch(AT)duke.poly.edu), Dec 11 2005
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