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

0,5

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).

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(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.

Example

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;

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 *)

Crossrefs

Cf. A182905, A001700.

Sequence in context: A136129 A278213 A034093 * A079508 A057150 A185663

Adjacent sequences: A246184 A246185 A246186 * A246188 A246189 A246190

Keyword

nonn,tabl

Author

Emeric Deutsch, Aug 29 2014

Status

approved