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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
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) = A182891(n,0).
G.f.: G(z) =1/( z^2+sqrt((1+z+z^2)*(1-3*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. Equivalently, a(n) ~ 5^(1/4) * phi^(2*n + 6) / (sqrt(Pi) * n^(3/2)), where phi = A001622 is the golden ratio. - Vaclav Kotesovec, Dec 06 2021
Conjecture: n*a(n) +(n-2)*a(n-1) +2*(-9*n+16)*a(n-2) +5*(2*n-5)*a(n-3) +(10*n-33) *a(n-4) +2*(26*n-109)*a(n-5) +(13*n-37)*a(n-6) +(13*n-63) *a(n-7) +10*(-n+7) *a(n-8)=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 H-steps at level 0.
MAPLE
G:=1/(z^2+sqrt((1+z+z^2)*(1-3*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)(1-3x+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)*(1-3*z+z^2)) )) \\ G. C. Greubel, Mar 26 2017
CROSSREFS
Cf. A182891.
Sequence in context: A077970 A338852 A174284 * A124696 A081669 A086821
KEYWORD
nonn
AUTHOR
Emeric Deutsch, Dec 12 2010
STATUS
approved

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Last modified March 28 18:04 EDT 2024. Contains 371254 sequences. (Running on oeis4.)