|
| |
|
|
A246180
|
|
Triangle read by rows: T(n,k) is the number of weighted lattice paths B(n) having k (1,0)-steps. 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 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, 0, 1, 0, 1, 1, 1, 0, 2, 1, 0, 3, 1, 3, 1, 0, 3, 6, 3, 4, 1, 2, 0, 12, 11, 6, 5, 1, 0, 10, 6, 30, 19, 10, 6, 1, 0, 10, 30, 30, 61, 31, 15, 7, 1, 5, 0, 60, 80, 90, 110, 48, 21, 8, 1, 0, 35, 30, 210, 200, 211, 183, 71, 28, 9, 1, 0, 35, 140, 210, 575, 462, 426, 287, 101, 36, 10, 1
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
0,9
|
|
|
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(3n,0)= A000108(n) (the Catalan numbers); T(n,0)=0 if n is not a multiple of 3.
|
|
|
LINKS
|
|
|
|
FORMULA
|
G.f. G=G(t,z) satisfies G = 1 + t*z*G + t*z^2*G + z^3*G^2.
|
|
|
EXAMPLE
|
Row 3 is 1,0,2,1. 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, 2, 2, and 3 (1,0)-steps, respectively.
Triangle starts:
1;
0,1;
0,1,1;
1,0,2,1;
0,3,1,3,1;
0,3,6,3,4,1;
|
|
|
MAPLE
|
eq := G = 1+t*z*G+t*z^2*G+z^3*G^2: G := RootOf(eq, G): Gser := simplify(series(G, z = 0, 17)): for n from 0 to 12 do P[n] := sort(coeff(Gser, z, n)) end do; for n from 0 to 12 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)*x+
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)):
|
|
|
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]*x + 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, 14}] // Flatten (* Jean-François Alcover, Jun 29 2015, after Alois P. Heinz *)
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
