A246187 Triangle read by rows: T(n,k) is the number of weighted lattice paths in F[n] having length k (length = number of steps). The members of F[n] are paths of weight n that start at (0,0), do not go below 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, 0, 1, 0, 2, 1, 0, 0, 5, 1, 0, 0, 4, 9, 1, 0, 0, 0, 17, 14, 1, 0, 0, 0, 8, 46, 20, 1, 0, 0, 0, 0, 49, 100, 27, 1, 0, 0, 0, 0, 16, 180, 190, 35, 1, 0, 0, 0, 0, 0, 129, 510, 329, 44, 1, 0, 0, 0, 0, 0, 32, 603, 1225, 532, 54, 1, 0, 0, 0, 0, 0, 0, 321, 2121, 2618, 816, 65, 1, 0, 0, 0, 0, 0, 0, 64, 1827, 6202, 5124, 1200, 77, 1

Offset: 0

Author

Emeric Deutsch, Aug 29 2014

Keywords

• nonn
• tabl

Comments

The paths need not end on the horizontal axis.

Number of entries in row n is n+1.

Sum of entries in row n is A182905(n).

Sum of entries in column k is A001700(k).

Examples

			Row 2 is 0,2,1; indeed, the weight-2 paths are hh, H, and U (where h=(1,0) of weight 1, H=(1,0) of weight 2, and U=(1,1)) and their lengths are 2,1,and 1, respectively.
Triangle starts:
1;
0,1;
0,2,1;
0,0,5,1;
0,0,4,9,1;
		

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.

Crossrefs

Cf. A182905, A001700.

Programs

Maple

eq := t*z^2*(1-2*t*z-2*t*z^2)*G^2+(1-t*z-3*t*z^2)*G-1 = 0: GG := RootOf(eq, G): Gser := simplify(series(GG, z = 0, 20)): for n from 0 to 18 do P[n] := sort(expand(coeff(Gser, z, n))) end do; for n from 0 to 18 do seq(coeff(P[n], t, j), j = 0 .. n) end do; # yields sequence in triangular form
# second Maple program:
b:= proc(n, y) option remember; `if`(y<0, 0,
      `if`(n=0, 1, expand(x*(b(n-1, y) +b(n-1, y-1)+
      `if`(n>1, b(n-2, y) +b(n-2, y+1), 0)))))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n, 0)):
seq(T(n), n=0..14);  # Alois P. Heinz, Aug 29 2014

Mathematica

b[n_, y_] := b[n, y] = If[y<0, 0, If[n==0, 1, Expand[x*(b[n-1, y] + b[n-1, y-1] + If[n>1, b[n-2, y] + b[n-2, y+1], 0])]]]; 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, Nov 07 2015, after Alois P. Heinz *)

Formula

G.f.: G(t,z) = g/(1-t*z^2*g), where g=g(t,z) satisfies g = 1 + t*z*g + t*z^2*g +t^2*z^3*g^2.

The g.f. G(t,z) satisfies t*z^2*(1 - 2*t*z - 2*t*z^2) + (1 - t*z - 3*t*z^2)*G - 1 = 0.