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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
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
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, N. Zagaglia Salvi, On the Rank Polynomial of the Lattice of Order Ideals of Fences and Crowns, Discrete Mathematics 259 (2002), 163-177.
FORMULA
a(n) = Sum_{k>=0} k*A182880(n,k).
G.f.: 2*z^3/[(1-3*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*n-5)*a(n-1) +(n-2)*a(n-2) +(2*n-3)*a(n-3) +(-n+1)*a(n-4)=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/((1-3*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/((1-3*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/((1-3*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
KEYWORD
nonn
AUTHOR
Emeric Deutsch, Dec 11 2010
STATUS
approved