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A102406
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Number of Dyck paths of semilength n having no ascents of length 1 that start at an even level.
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2
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1, 0, 1, 2, 5, 14, 39, 114, 339, 1028, 3163, 9852, 31005, 98436, 314901, 1014070, 3284657, 10694314, 34979667, 114887846, 378750951, 1252865288, 4157150327, 13832926200, 46148704121, 154327715592, 517236429545, 1737102081962
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,4
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COMMENTS
| Number of Lukasiewicz paths of length n having no level steps at an even level. A Lukasiewicz path of length n is a path in the first quadrant from (0,0) to (n,0) using rise steps (1,k) for any positive integer k, level steps (1,0) and fall steps (1,-1) (see R. P. Stanley, Enumerative Combinatorics, Vol. 2, Cambridge Univ. Press, Cambridge, 1999, p. 223, Exercise 6.19w; the integers are the slopes of the steps). Example: a(3)=2 because we have UHD and U(2)DD, where U=(1,1), H=(1,0), D=(1,-1) and U(2)=(1,2). a(n)=A102404(n,0).
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FORMULA
| G.f.=[1+z+z^2-sqrt(1-2z-5z^2-2z^3+z^4)]/[2z(1+z)^2]
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EXAMPLE
| a(3)=2 because we have UUDUDD and UUUDDD, having no ascents of length 1 that start at an even level.
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MAPLE
| G:=(1+z+z^2-sqrt(1-2*z-5*z^2-2*z^3+z^4))/2/z/(1+z)^2: Gser:=series(G, z=0, 32): 1, seq(coeff(Gser, z^n), n=1..29);
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CROSSREFS
| Cf. A102404, A102407.
Sequence in context: A201778 A105641 A027035 * A151409 A003054 A148316
Adjacent sequences: A102403 A102404 A102405 * A102407 A102408 A102409
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KEYWORD
| nonn
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AUTHOR
| Emeric Deutsch (deutsch(AT)duke.poly.edu), Jan 06 2005
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