A182895
Number of (1,0)-steps at level 0 in all weighted lattice paths in L_n.
Original entry on oeis.org
0, 1, 3, 7, 19, 50, 130, 341, 893, 2337, 6119, 16020, 41940, 109801, 287463, 752587, 1970299, 5158310, 13504630, 35355581, 92562113, 242330757, 634430159, 1660959720, 4348449000, 11384387281, 29804712843, 78029751247, 204284540899
Offset: 0
a(3) = 7. 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; they contain 0+0+2+2+3=7 (1,0)-steps at level 0.
- 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.
- Index entries for linear recurrences with constant coefficients, signature (2,1,2,-1)
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G:=z*(1+z)/(1+z+z^2)/(1-3*z+z^2): Gser:=series(G,z=0,32): seq(coeff(Gser,z,n),n=0..28);
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LinearRecurrence[{2,1,2,-1},{0,1,3,7},30] (* Harvey P. Dale, Jan 05 2022 *)
A182894
Number of weighted lattice paths in L_n having no (1,0)-steps at level 0. 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.
Original entry on oeis.org
1, 0, 0, 2, 2, 4, 12, 24, 54, 130, 300, 706, 1686, 4028, 9686, 23426, 56866, 138584, 338940, 831508, 2045736, 5046240, 12477290, 30919122, 76774382, 190995224, 475979602, 1188125394, 2970282794, 7436232760, 18641883396, 46792219972, 117590713254
Offset: 0
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; two of them, namely ud and du, have no (1,0)-steps at level 0.
- 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.
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G:=1/(z+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..32);
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CoefficientList[Series[1/(x+x^2+Sqrt[(1+x+x^2)*(1-3*x+x^2)]), {x, 0, 20}], x] (* Vaclav Kotesovec, Mar 06 2016 *)
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