

A182889


Number of weighted lattice paths in L_n having no (1,0)steps at level 0. 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.


1



1, 0, 1, 2, 3, 8, 17, 38, 89, 206, 485, 1152, 2751, 6614, 15983, 38798, 94569, 231342, 567771, 1397562, 3449285, 8533886, 21161001, 52579900, 130896887, 326440746, 815437967, 2040049514, 5111051473, 12822135138, 32207384995, 80995950182, 203917464635
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

0,4


COMMENTS

a(n)=A182888(n,0).


REFERENCES

M. Bona and A. Knopfmacher, On the probability that certain compositions have the same number of parts, Ann. Comb., 14 (2010), 291306.
E. Munarini, N. Zagaglia Salvi, On the rank polynomial of the lattice of order ideals of fences and crowns, Discrete Mathematics 259 (2002), 163177.


LINKS

Table of n, a(n) for n=0..32.


FORMULA

G.f.: G(z) =1/( z+sqrt((1+z+z^2)(13z+z^2)) ).
a(n) ~ sqrt(7*sqrt(5)15) * ((3 + sqrt(5))/2)^(n+2) / (sqrt(2*Pi) * n^(3/2)).  Vaclav Kotesovec, Mar 06 2016


EXAMPLE

a(3)=2. Indeed, 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; two of them, namely ud and du, have no h steps at level 0.


MAPLE

G:=1/(z+sqrt((1+z+z^2)*(13*z+z^2))): Gser:=series(G, z=0, 35): seq(coeff(Gser, z, n), n=0..32);


MATHEMATICA

CoefficientList[Series[1/(x+Sqrt[(1+x+x^2)(13x+x^2)]), {x, 0, 40}], x] (* Harvey P. Dale, Jun 16 2013 *)


CROSSREFS

Cf. A182888.
Sequence in context: A267223 A292401 A132333 * A256169 A298405 A219788
Adjacent sequences: A182886 A182887 A182888 * A182890 A182891 A182892


KEYWORD

nonn


AUTHOR

Emeric Deutsch, Dec 11 2010


STATUS

approved



