

A182881


Number of (1,1)steps in all weighted lattice paths in L_n.


2



0, 0, 0, 2, 6, 18, 56, 162, 462, 1306, 3648, 10116, 27892, 76524, 209112, 569506, 1546542, 4189314, 11323480, 30548190, 82272330, 221240070, 594131160, 1593553452, 4269391596, 11426761548, 30554523096, 81631135502, 217918012002
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OFFSET

0,4


COMMENTS

L_n is the set of lattice 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; 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) = Sum_{k>=0} k*A182880(n,k).
G.f.: 2*z^3/[(13*z+z^2)*(1+z+z^2)]^(3/2).
a(n) ~ ((3 + sqrt(5))/2)^n * sqrt(n) / (2*sqrt(Pi)*5^(3/4)).  Vaclav Kotesovec, Mar 06 2016
Conjecture: (n+3)*a(n) +(2*n5)*a(n1) +(n2)*a(n2) +(2*n3)*a(n3) +(n+1)*a(n4)=0.  R. J. Mathar, Jun 14 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, containing a total of 1+1+0+0+0=2 u steps.


MAPLE

g:=2*z^3/((13*z+z^2)*(1+z+z^2))^(3/2): gser:=series(g, z=0, 32): seq(coeff(gser, z, n), n=0..28);


MATHEMATICA

CoefficientList[Series[2*x^3/((13*x+x^2)*(1+x+x^2))^(3/2), {x, 0, 20}], x] (* Vaclav Kotesovec, Mar 06 2016 *)


PROG

(PARI) z='z+O('z^50); concat([0, 0, 0], Vec(2*z^3/((13*z+z^2)*(1+z+z^2))^(3/2))) \\ G. C. Greubel, Mar 25 2017


CROSSREFS

Cf. A182880.
Sequence in context: A066158 A148456 A148457 * A291730 A002999 A291228
Adjacent sequences: A182878 A182879 A182880 * A182882 A182883 A182884


KEYWORD

nonn


AUTHOR

Emeric Deutsch, Dec 11 2010


STATUS

approved



