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 A182898 Triangle read by rows: T(n,k) is the number of weighted lattice paths in L_n having k returns to the horizontal axis (both from above and below). 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. 1
 1, 1, 2, 3, 2, 5, 6, 8, 18, 13, 46, 4, 21, 112, 20, 34, 262, 80, 55, 600, 268, 8, 89, 1356, 816, 56, 144, 3046, 2324, 280, 233, 6832, 6320, 1144, 16, 377, 15354, 16620, 4136, 144, 610, 34658, 42652, 13728, 864, 987, 78706, 107520, 42816, 4144, 32, 1597, 180000, 267564, 127392, 17264, 352 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS Sum of entries in row n is A051286(n). T(n,0)=A000045(n+1) (the Fibonacci numbers). Sum(k*T(n,k), k=0..n)=A182899(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 Table of n, a(n) for n=0..56. FORMULA G.f.: G(t,z) =1/[1-z-z^2-2tz^3*c], where c satisfies c = 1+zc+z^2*c+z^3*c^2. The trivariate g.f. H=H(t,s,z), where t (s) marks (1,-1)-returns ((1,1)-returns) to the horizontal axis, and z marks weight is given by H=1+zH+z^2*H+(t+s)z^3*cH, where c satisfies c = 1+zc+z^2*c+z^3*c^2. EXAMPLE T(3,1)=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 1 return to the horizontal axis. Triangle starts: 1; 1; 2; 3,2; 5,6; 8,18; 13,46,4; 21,112,20; MAPLE eq := c = 1+z*c+z^2*c+z^3*c^2: c := RootOf(eq, c): G := 1/(1-z-z^2-2*t*z^3*c): Gser := simplify(series(G, z = 0, 20)): for n from 0 to 16 do P[n] := sort(coeff(Gser, z, n)) end do: for n from 0 to 16 do seq(coeff(P[n], t, k), k = 0 .. floor((1/3)*n)) end do; # yields sequence in triangular form CROSSREFS Sequence in context: A241592 A211506 A182880 * A133684 A286151 A192138 Adjacent sequences: A182895 A182896 A182897 * A182899 A182900 A182901 KEYWORD nonn,tabf AUTHOR Emeric Deutsch, Dec 13 2010 STATUS approved

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Last modified December 7 01:11 EST 2023. Contains 367616 sequences. (Running on oeis4.)