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 A182882 Triangle read by rows: T(n,k) is the number of weighted lattice paths in L_n having k (1,0)-steps of weight 1. L_n is the set of lattice 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; 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, 1, 0, 1, 2, 2, 0, 1, 1, 6, 3, 0, 1, 6, 3, 12, 4, 0, 1, 7, 24, 6, 20, 5, 0, 1, 12, 34, 60, 10, 30, 6, 0, 1, 31, 60, 100, 120, 15, 42, 7, 0, 1, 40, 185, 180, 230, 210, 21, 56, 8, 0, 1, 91, 260, 645, 420, 455, 336, 28, 72, 9, 0, 1, 170, 636, 980, 1715, 840, 812, 504, 36, 90, 10, 0, 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)=A182883(n). Sum(k*T(n,k), k=0..n)=A182884(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/sqrt(1-2tz-2z^2+t^2*z^2+2t*z^3+z^4-4z^3). 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 have exactly one h step. Triangle starts: 1; 0,1; 1,0,1; 2,2,0,1; 1,6,3,0,1; 6,3,12,4,0,1 MAPLE G:=1/sqrt(1-2*t*z-2*z^2+t^2*z^2+2*t*z^3+z^4-4*z^3): 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, A182883, A182884. Sequence in context: A264909 A104579 A079531 * A134178 A059018 A249371 Adjacent sequences:  A182879 A182880 A182881 * A182883 A182884 A182885 KEYWORD nonn,tabl AUTHOR Emeric Deutsch, Dec 11 2010 STATUS approved

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Last modified May 12 22:25 EDT 2021. Contains 343829 sequences. (Running on oeis4.)