OFFSET
0,7
COMMENTS
A Motzkin path of length n has n+1 nodes.
LINKS
Alois P. Heinz, Rows n = 0..28, flattened
Wikipedia, Motzkin number
EXAMPLE
In the A001006(3) = 4 Motzkin paths of length 3 there are 2 levels with 1 node, 2 levels with 2 nodes, 2 levels with 3 nodes, and 1 level with 4 nodes.
2 _ 1 1
2 / \ 3 /\_ 3 _/\ 4 ___ .
So row 3 is [2, 2, 2, 1].
Triangle T(n,k) begins:
1;
0, 1;
1, 1, 1;
2, 2, 2, 1;
4, 6, 4, 3, 1;
8, 14, 12, 7, 4, 1;
18, 32, 33, 21, 11, 5, 1;
44, 74, 84, 64, 34, 16, 6, 1;
113, 180, 208, 181, 111, 52, 22, 7, 1;
296, 457, 520, 485, 344, 179, 76, 29, 8, 1;
782, 1195, 1334, 1273, 1000, 599, 274, 107, 37, 9, 1;
...
MAPLE
g:= proc(x, y, p) (h-> `if`(x=0, add(z^coeff(h, z, i)
, i=0..degree(h)), b(x, y, h)))(p+z^y) end:
b:= proc(x, y, p) option remember; `if`(y+2<=x, g(x-1, y+1, p), 0)
+`if`(y+1<=x, g(x-1, y, p), 0)+`if`(y>0, g(x-1, y-1, p), 0)
end:
T:= n-> (p-> seq(coeff(p, z, i), i=1..n+1))(g(n, 0$2)):
seq(T(n), n=0..10);
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Alois P. Heinz, Apr 15 2024
STATUS
approved