OFFSET
0,3
COMMENTS
L_n is the set of lattice paths of weight n that start at (0,0) and end on the horizontal axis and whose steps are of the following four kinds: a (1,0)-step with weight 1, a (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
Jean-Luc Baril, Nathanaël Hassler, Sergey Kirgizov, and José L. Ramírez, Grand zigzag knight's paths, arXiv:2402.04851 [math.CO], 2024.
Miklós Bóna and Arnold Knopfmacher, On the probability that certain compositions have the same number of parts, Ann. Comb., 14 (2010), 291-306.
Emanuele Munarini and Norma 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..n} k*binomial(k,n-k)^2.
G.f.: z*(1+2*z^2-z^3)/((1-3*z+z^2)*(1+z+z^2))^(3/2).
a(n) ~ sqrt(20 + 9*sqrt(5)) * ((3 + sqrt(5))/2)^n * sqrt(n) / (10*sqrt(Pi)). - Vaclav Kotesovec, Mar 06 2016
Conjecture: (n-1)*(310*n-781)*a(n) + (-882*n^2+3845*n-3679)*a(n-1) + (214*n^2-2903*n+3751)*a(n-2) + (-358*n^2-459*n+1955)*a(n-3) + (834*n^2-3631*n+4065)*a(n-4) - (262*n-663)*(n-3)*a(n-5) = 0. - R. J. Mathar, Jun 14 2016
Recurrence (of order 4): (n-1)*(2*n-5)*(2*n^2 - 15*n + 24)*a(n) = 2*(n-3)*(4*n^3 - 30*n^2 + 52*n - 13)*a(n-1) + (4*n^4 - 40*n^3 + 133*n^2 - 201*n + 96)*a(n-2) + 2*(4*n^4 - 38*n^3 + 110*n^2 - 126*n + 51)*a(n-3) - (n-2)*(2*n-3)*(2*n^2 - 11*n + 11)*a(n-4). - Vaclav Kotesovec, Sep 23 2017
a(n) ~ phi^(2*n + 3) * sqrt(n) / (2 * 5^(3/4) * sqrt(Pi)), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Sep 23 2017
EXAMPLE
a(3)=11. Indeed, denoting by h (H) the (1,0)-step of weight 1 (2), and u=(1,1), d=(1,-1), there are five paths of weight 3: hhh, hH, Hh, ud, and du; their lengths are 3, 2, 2, 2,and 2, respectively.
MAPLE
a:=n->sum(k*binomial(k, n-k)^2, k=0..n): seq(a(n), n=0..28);
MATHEMATICA
CoefficientList[Series[x*(1+2*x^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], Vec(z*(1+2*z^2-z^3)/((1-3*z+z^2)*(1+z+z^2))^(3/2))) \\ G. C. Greubel, Mar 25 2017
CROSSREFS
KEYWORD
nonn
AUTHOR
Emeric Deutsch, Dec 11 2010
STATUS
approved