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A182899
Number of returns to the horizontal axis (both from above and below) in all weighted lattice paths in L_n.
1
0, 0, 0, 2, 6, 18, 54, 152, 422, 1160, 3156, 8534, 22968, 61578, 164602, 438930, 1168120, 3103540, 8234122, 21820098, 57762774, 152774358, 403750258, 1066291206, 2814322014, 7423962336, 19574314938, 51587866820, 135905559330, 357908155044
OFFSET
0,4
COMMENTS
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.
LINKS
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 and N. Zagaglia Salvi, On the Rank Polynomial of the Lattice of Order Ideals of Fences and Crowns, Discrete Mathematics 259 (2002), 163-177.
FORMULA
a(n) = Sum_{k>=0} k*A182898(n,k).
a(n) = 2*A182897(n).
G.f.: 2*z^3*c/((1+z+z^2)*(1-3*z+z^2)), where c satisfies c = 1+z*c+z^2*c+z^3*c^2.
Conjecture D-finite with recurrence n*a(n) +(-4*n+3)*a(n-1) +(2*n-3)*a(n-2) +11*(n-3)*a(n-4) +(2*n-9)*a(n-6) +(-4*n+21)*a(n-7) +(n-6)*a(n-8)=0. - R. J. Mathar, Jul 22 2022
EXAMPLE
a(3)=2 because, denoting by h (H) the (1,0)-step of weight 1 (2), and u=(1,1), d=(1,-1), the five paths of weight 3 are ud, du, hH, Hh, and hhh; they contain 1+1+0+0+0=1 returns to the horizontal axis.
MAPLE
eq := c = 1+z*c+z^2*c+z^3*c^2: c := RootOf(eq, c): G := 2*z^3*c/((1+z+z^2)*(1-3*z+z^2)): Gser := series(G, z = 0, 32): seq(coeff(Gser, z, n), n = 0 .. 29);
CROSSREFS
KEYWORD
nonn
AUTHOR
Emeric Deutsch, Dec 13 2010
STATUS
approved