OFFSET
0,9
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)
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 = 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.
EXAMPLE
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;
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 *)
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Emeric Deutsch, Sep 17 2014
STATUS
approved