

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
(list;
graph;
refs;
listen;
history;
text;
internal format)



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



FORMULA

G.f.: 2*z^3*c/((1+z+z^2)*(13*z+z^2)), where c satisfies c = 1+z*c+z^2*c+z^3*c^2.
Conjecture Dfinite with recurrence n*a(n) +(4*n+3)*a(n1) +(2*n3)*a(n2) +11*(n3)*a(n4) +(2*n9)*a(n6) +(4*n+21)*a(n7) +(n6)*a(n8)=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)*(13*z+z^2)): Gser := series(G, z = 0, 32): seq(coeff(Gser, z, n), n = 0 .. 29);


CROSSREFS



KEYWORD

nonn


AUTHOR



STATUS

approved



