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.

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

Offset: 0

Author

Emeric Deutsch, Sep 17 2014

Keywords

• nonn
• tabl

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)

Examples

			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;
		

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. A004148, A166300, A247300,

Programs

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 *)

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.