

A182892


Number of weighted lattice paths in L_n having no (1,0)steps of weight 2 at level 0.


2



1, 1, 1, 3, 7, 15, 35, 83, 197, 473, 1145, 2787, 6819, 16759, 41345, 102341, 254075, 632437, 1577967, 3945517, 9884379, 24806201, 62355121, 156974319, 395712759, 998809135, 2524043569, 6385400005, 16170553755, 40990092629, 103997889735
(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

G. C. Greubel, Table of n, a(n) for n = 0..1000
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.


FORMULA

a(n) = A182891(n,0).
G.f.: G(z) =1/( z^2+sqrt((1+z+z^2)*(13*z+z^2)) ).
a(n) ~ sqrt(360 + 161*sqrt(5)) * ((3 + sqrt(5))/2)^n / (sqrt(Pi)*n^(3/2)).  Vaclav Kotesovec, Mar 06 2016
Conjecture: n*a(n) +(n2)*a(n1) +2*(9*n+16)*a(n2) +5*(2*n5)*a(n3) +(10*n33) *a(n4) +2*(26*n109)*a(n5) +(13*n37)*a(n6) +(13*n63) *a(n7) +10*(n+7) *a(n8)=0.  R. J. Mathar, Jun 14 2016


EXAMPLE

a(3)=3. 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; three of them, namely ud, du, and hhh, have no Hsteps at level 0.


MAPLE

G:=1/(z^2+sqrt((1+z+z^2)*(13*z+z^2))): Gser:=series(G, z=0, 35): seq(coeff(Gser, z, n), n=0..30);


MATHEMATICA

CoefficientList[Series[1/(x^2+Sqrt[(1+x+x^2)(13x+x^2)]), {x, 0, 30}], x] (* Harvey P. Dale, Aug 25 2012 *)


PROG

(PARI) z='z+O('z^50); Vec(1/( z^2+sqrt((1+z+z^2)*(13*z+z^2)) )) \\ G. C. Greubel, Mar 26 2017


CROSSREFS

Cf. A182891.
Sequence in context: A077970 A338852 A174284 * A124696 A081669 A086821
Adjacent sequences: A182889 A182890 A182891 * A182893 A182894 A182895


KEYWORD

nonn


AUTHOR

Emeric Deutsch, Dec 12 2010


STATUS

approved



