A182892 Number of weighted lattice paths in L_n having no (1,0)-steps of weight 2 at level 0.
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
Offset: 0
Keywords
Examples
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.
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), 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.
Crossrefs
Cf. A182891.
Programs
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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);
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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 *) -
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
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
Comments