A246179 Triangle read by rows: T(n,k) is the number of weighted lattice paths in B(n) having k returns to the horizontal axis (i.e., (1,-1)-steps ending on the horizontal axis). The members of B(n) are paths of weight n that start in (0,0), end on but never go below the horizontal axis, and whose steps are of the following four kinds: a (1,0)-step with weight 1; a (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.

1, 1, 2, 3, 1, 5, 3, 8, 9, 13, 23, 1, 21, 56, 5, 34, 131, 20, 55, 300, 67, 1, 89, 678, 204, 7, 144, 1523, 581, 35, 233, 3416, 1580, 143, 1, 377, 7677, 4155, 517, 9, 610, 17329, 10663, 1716, 54, 987, 39353, 26880, 5352, 259, 1, 1597, 90000, 66891, 15924, 1079

Offset: 0

Author

Emeric Deutsch, Aug 23 2014

Keywords

• nonn
• tabf

Comments

Number of entries in row n is 1+floor(n/3).

Sum of entries in row n is A004148(n+1) (the 2ndary structure numbers).

T(n,0)=A000045(n+1) (the Fibonacci numbers).

Examples

			Row 3 is 3,1. Indeed, denoting by h (H) the (1,0)-step of weight 1 (2), and u=(1,1), d=(1,-1), the four paths of weight 3 are: ud, hH, Hh, and hhh, having 1, 0, 0, and 0 returns to the horizontal axis, respectively.
Triangle starts:
1;
1;
2;
3,1;
5,3;
8,9;
13,23,1;
		

Links

• Alois P. Heinz, Rows n = 0..250, 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, A000045, A246177.

Programs

Maple

eq := g = 1+z*g+z^2*g+z^3*g^2: g := RootOf(eq, g): G := 1/(1-z-z^2-t*z^3*g): Gser := simplify(series(G, z = 0, 20)): for n from 0 to 18 do P[n] := sort(coeff(Gser, z, n)) end do: for n from 0 to 18 do seq(coeff(P[n], t, k), k = 0 .. floor((1/3)*n)) end do; # yields sequence in triangular form
# second Maple program:
b:= proc(n, y) option remember; `if`(y<0 or y>n, 0,
      `if`(n=0, 1, expand(b(n-1, y)+`if`(n>1, b(n-2, y)+
       b(n-2, y+1), 0) +b(n-1, y-1)*`if`(y=1, x, 1))))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, 0)):
seq(T(n), n=0..20); # Alois P. Heinz, Aug 24 2014

Mathematica

b[n_, y_] := b[n, y] = If[y<0 || y>n, 0, If[n==0, 1, Expand[b[n-1, y] + If[n>1, b[n-2, y] + b[n-2, y+1], 0] + b[n-1, y-1]*If[y==1, x, 1]]]]; T[n_] := Function[ {p}, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][b[n, 0]]; Table[T[n], {n, 0, 20}] // Flatten (* Jean-François Alcover, Jun 29 2015, after Alois P. Heinz *)

Formula

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