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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, 5845077156189, 19702791805126
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OFFSET
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0,4
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COMMENTS
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Number of Łukasiewicz paths of length n having no level steps at an even level. A Łukasiewicz 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).
Number of Dyck n-paths with no descent of length 1 following an ascent of length 1. [David Scambler, May 11 2012]
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LINKS
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FORMULA
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G.f.: [1+z+z^2-sqrt(1-2z-5z^2-2z^3+z^4)]/[2z(1+z)^2].
Conjecture: (n+1)*a(n) +(-n+3)*a(n-1) +(-7*n+9)*a(n-2) +(-7*n+12)*a(n-3) -n*a(n-4) +(n-4)*a(n-5)=0. - R. J. Mathar, Jan 04 2017
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EXAMPLE
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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
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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
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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