A246181 Triangle read by rows: T(n,k) is the number of weighted lattice paths B(n) having k (1,0)-steps of weight 1. 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: a (1,0)-step of weight 1; a (1,0)-step of weight 2; a (1,1)-step of weight 2; a (1,-1)-step of weight 1. The weight of a path is the sum of the weights of its steps.

1, 0, 1, 1, 0, 1, 1, 2, 0, 1, 1, 3, 3, 0, 1, 3, 3, 6, 4, 0, 1, 3, 12, 6, 10, 5, 0, 1, 6, 14, 30, 10, 15, 6, 0, 1, 11, 30, 40, 60, 15, 21, 7, 0, 1, 15, 65, 90, 90, 105, 21, 28, 8, 0, 1, 31, 95, 225, 210, 175, 168, 28, 36, 9, 0, 1, 50, 216, 350, 595, 420, 308, 252, 36, 45, 10, 0, 1

Offset: 0

Author

Emeric Deutsch, Aug 23 2014

Keywords

• nonn
• tabl

Comments

Number of entries in row n is n+1.

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

T(n,0) = A025250(n+3).

Sum(k*T(n,k), k>=0) = A110320(n) (n>=1).

Examples

			Row 3 is 1,2,0,1. 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 0, 1, 1, and 3 (1,0)-steps of weight 1, respectively.
Triangle starts:
1;
0,1;
1,0,1;
1,2,0,1;
1,3,3,0,1;
3,3,6,4,0,1;
		

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, A025250, A246177.

Programs

Maple

eq := G = 1+t*z*G+z^2*G+z^3*G^2: G := RootOf(eq, G): Gser := simplify(series(G, z = 0, 22)): for n from 0 to 17 do P[n] := sort(coeff(Gser, z, n)) end do: for n from 0 to 15 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, 0,
      `if`(n=0, 1, expand(b(n-1, y)*x+ `if`(n>1,
       b(n-2, y)+b(n-2, y+1), 0) +b(n-1, y-1))))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, 0)):
seq(T(n), n=0..12); # 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]*x + If[n>1, b[n-2, y] + b[n-2, y+1], 0] + b[n-1, y-1]]]]; T[n_] := Function[{p}, Table[ Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][b[n, 0]]; Table[T[n], {n, 0, 12}] // Flatten (* Jean-François Alcover, Jun 29 2015, after Alois P. Heinz *)

Formula

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