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

0,3

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).

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.

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.

Example

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;

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 *)

Crossrefs

Cf. A004148, A000045, A246177.

Sequence in context: A352525 A114711 A246186 * A166285 A336365 A002472

Adjacent sequences: A246176 A246177 A246178 * A246180 A246181 A246182

Keyword

nonn,tabf

Author

Emeric Deutsch, Aug 23 2014

Status

approved