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 A247299 Triangle read by rows: T(n,k) is the number of weighted lattice paths B(n) having a total of k h- and H-steps at level 0. 2
 1, 0, 1, 0, 1, 1, 1, 0, 2, 1, 1, 2, 1, 3, 1, 2, 4, 3, 3, 4, 1, 5, 6, 9, 5, 6, 5, 1, 10, 15, 15, 16, 9, 10, 6, 1, 22, 33, 33, 32, 26, 16, 15, 7, 1, 50, 71, 78, 66, 60, 41, 27, 21, 8, 1, 113, 163, 171, 158, 125, 103, 64, 43, 28, 9, 1, 260, 374, 391, 360, 295, 225, 167, 99, 65, 36, 10, 1 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,9 COMMENTS B(n) is the set of lattice paths of weight n that start in (0,0), end on the horizontal axis and never go below this axis, whose steps are of the following four kinds: h = (1,0) of weight 1, H = (1,0) of weight 2, u = (1,1) of weight 2, and d = (1,-1) of weight 1. The weight of a path is the sum of the weights of its steps. Row n contains n+1 entries. Sum of entries in row n is A004148(n+1) (the 2ndary structure numbers). T(n,0) = A166300(n). Sum(k*T(n,k), k=0..n) = A247300(n) LINKS Alois P. Heinz, Rows n = 0..140, flattened M. Bona and A. Knopfmacher, On the probability that certain compositions have the same number of parts, Ann. Comb., 14 (2010), 291-306. FORMULA G.f. G = 1/(1 - t*z - t*z^2 - z^3*g), where g is given by g = 1 + z*g + z^2*g + z^3*g^2. EXAMPLE Row 3 is 1,0,2,1 because B(3) = {ud, hH, Hh, hhh}. Triangle starts: 1; 0,1; 0,1,1; 1,0,2,1; 1,2,1,3,1; 2,4,3,3,4,1; MAPLE eqg := g = 1+z*g+z^2*g+z^3*g^2: g := RootOf(eqg, g): G := 1/(1-t*z-t*z^2-z^3*g): Gser := simplify(series(G, z = 0, 16)): for n from 0 to 13 do P[n] := sort(coeff(Gser, z, n)) end do: for n from 0 to 13 do seq(coeff(P[n], t, k), k = 0 .. n) end do; # yields sequence in triangular form # second Maple program: b:= proc(n, y) option remember; `if`(y<0 or y>n or n<0, 0, `if`(n=0, 1, expand(`if`(y=0, x, 1)*(b(n-1, y)+ b(n-2, y)) +b(n-2, y+1) +b(n-1, y-1)))) end: T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n, 0)): seq(T(n), n=0..14); # Alois P. Heinz, Sep 17 2014 MATHEMATICA b[n_, y_] := b[n, y] = If[y<0 || y>n || n<0, 0, If[n == 0, 1, Expand[If[y == 0, x, 1]*(b[n-1, y] + b[n-2, y]) + b[n-2, y+1] + b[n-1, y-1]]]]; T[n_] := Function[{p}, Table[Coefficient[p, x, i], {i, 0, n}]][b[n, 0]]; Table[T[n], {n, 0, 14}] // Flatten (* Jean-François Alcover, Feb 19 2015, after Alois P. Heinz *) CROSSREFS Cf. A004148, A166300, A247300, Sequence in context: A212182 A236567 A360564 * A127586 A055893 A050221 Adjacent sequences: A247296 A247297 A247298 * A247300 A247301 A247302 KEYWORD nonn,tabl AUTHOR Emeric Deutsch, Sep 17 2014 STATUS approved

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Last modified March 5 05:51 EST 2024. Contains 370537 sequences. (Running on oeis4.)