A182899 Number of returns to the horizontal axis (both from above and below) in all weighted lattice paths in L_n.

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

Author

Emeric Deutsch, Dec 13 2010

Keywords

• nonn

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.

Examples

			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.
		

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.

Crossrefs

Cf. A182896, A182897, A182898.

Programs

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);

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