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 A182893 Triangle read by rows: T(n,k) is the number of weighted lattice paths in L_n having k (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. 2
 1, 0, 1, 0, 1, 1, 2, 0, 2, 1, 2, 4, 1, 3, 1, 4, 8, 6, 3, 4, 1, 12, 12, 18, 9, 6, 5, 1, 24, 36, 30, 32, 14, 10, 6, 1, 54, 84, 78, 64, 51, 22, 15, 7, 1, 130, 184, 204, 152, 120, 77, 34, 21, 8, 1, 300, 452, 462, 416, 280, 205, 113, 51, 28, 9, 1, 706, 1084, 1130, 1000, 770, 492, 328, 163, 74, 36, 10, 1 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,7 COMMENTS Sum of entries in row n is A051286(n). T(n,0)=A182894(n). Sum(k*T(n,k), k=0..n)=A182895(n). REFERENCES 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. LINKS FORMULA G.f.: G(t,z) =1/[(1-t)z(1+z)+sqrt((1+z+z^2)(1-3z+z^2))]. EXAMPLE T(3,2)=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 hH and Hh, have exactly two (1,0)-steps at level 0. Triangle starts: 1; 0,1; 0,1,1; 2,0,2,1; 2,4,1,3,1; 4,8,6,3,4,1. MAPLE G:=1/((1-t)*z*(1+z)+sqrt((1+z+z^2)*(1-3*z+z^2))): Gser:=simplify(series(G, z=0, 15)): for n from 0 to 11 do P[n]:=sort(coeff(Gser, z, n)) od: for n from 0 to 11 do seq(coeff(P[n], t, k), k=0..n) od; # yields sequence in triangular form CROSSREFS Cf. A051286, A182894, A182895. Sequence in context: A334305 A125922 A283306 * A206298 A076608 A068461 Adjacent sequences:  A182890 A182891 A182892 * A182894 A182895 A182896 KEYWORD nonn,tabl AUTHOR Emeric Deutsch, Dec 12 2010 STATUS approved

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Last modified April 21 02:10 EDT 2021. Contains 343143 sequences. (Running on oeis4.)