A182904 Number of weighted lattice paths in L_n having no peaks. The members of L_n are paths of weight n that start at (0,0), end on the horizontal axis and whose steps are of the following four kinds: an (1,0)-step with weight 1, an (1,0)-step with weight 2, a (1,1)-step with weight 2, and a (1,-1)-step with weight 1. The weight of a path is the sum of the weights of its steps. A peak is a (1,1)-step followed by a (1,-1)-step.
1, 1, 2, 4, 9, 21, 48, 112, 263, 623, 1484, 3550, 8525, 20537, 49612, 120136, 291519, 708699, 1725714, 4208364, 10276173, 25122829, 61486180, 150632012, 369361757, 906462529, 2226297008, 5471757126, 13457326605, 33117622245, 81547372396
Offset: 0
Keywords
Examples
a(3)=4. Indeed, denoting by h (H) the (1,0)-step of weight 1 (2), and U=(1,1), D=(1,-1), we have hhh, hH, Hh, and DU.
References
- M. Bona and A. Knopfmacher, On the probability that certain compositions have the same number of parts, Ann. Comb., 14 (2010), 291-306.
- E. Munarini, N. Zagaglia Salvi, On the rank polynomial of the lattice of order ideals of fences and crowns, Discrete Mathematics 259 (2002), 163-177.
Crossrefs
Cf. A182903.
Programs
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Maple
g := 1/sqrt(1-2*z-z^2-z^4-2*z^5+z^6): gser := series(g, z = 0, 35):seq(coeff(gser, z, n), n = 0 .. 30);
Formula
G.f.: g=1/sqrt(1-2z-z^2-z^4-2z^5+z^6).
Conjecture: n*a(n) +(n-2)*a(n-1) +(-7*n+10)*a(n-2) +3*(-n+2)*a(n-3) +(-n+2)*a(n-4) +(-5*n+14)*a(n-5) +(-5*n+18)*a(n-6) +3*(n-4)*a(n-7)=0. - R. J. Mathar, Jun 14 2016
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