

A247299


Triangle read by rows: T(n,k) is the number of weighted lattice paths B(n) having a total of k h and Hsteps at level 0.


2



1, 0, 1, 0, 1, 1, 1, 0, 2, 1, 1, 2, 1, 3, 1, 2, 4, 3, 3, 4, 1, 5, 6, 9, 5, 6, 5, 1, 10, 15, 15, 16, 9, 10, 6, 1, 22, 33, 33, 32, 26, 16, 15, 7, 1, 50, 71, 78, 66, 60, 41, 27, 21, 8, 1, 113, 163, 171, 158, 125, 103, 64, 43, 28, 9, 1, 260, 374, 391, 360, 295, 225, 167, 99, 65, 36, 10, 1
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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).


LINKS



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/(1t*zt*z^2z^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(n1, y)+
b(n2, y)) +b(n2, y+1) +b(n1, y1))))
end:
T:= n> (p> seq(coeff(p, x, i), i=0..n))(b(n, 0)):


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[n1, y] + b[n2, y]) + b[n2, y+1] + b[n1, y1]]]]; T[n_] := Function[{p}, Table[Coefficient[p, x, i], {i, 0, n}]][b[n, 0]]; Table[T[n], {n, 0, 14}] // Flatten (* JeanFrançois Alcover, Feb 19 2015, after Alois P. Heinz *)


CROSSREFS



KEYWORD



AUTHOR



STATUS

approved



