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

0,8

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).

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

Example

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;

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 *)

Crossrefs

Cf. A004148, A025250, A246177.

Sequence in context: A097559 A293108 A172237 * A123226 A347382 A299919

Adjacent sequences: A246178 A246179 A246180 * A246182 A246183 A246184

Keyword

nonn,tabl

Author

Emeric Deutsch, Aug 23 2014

Status

approved