A247622 Triangular array: T(n,k) = number of paths from (0,0) to (n,k), each segment given by a vector (1,1), (1,-1), or (2,0), not crossing the x-axis, and including no horizontal segment on the x-axis.

1, 0, 1, 1, 0, 1, 0, 3, 0, 1, 3, 0, 5, 0, 1, 0, 11, 0, 7, 0, 1, 11, 0, 23, 0, 9, 0, 1, 0, 45, 0, 39, 0, 11, 0, 1, 45, 0, 107, 0, 59, 0, 13, 0, 1, 0, 197, 0, 205, 0, 83, 0, 15, 0, 1, 197, 0, 509, 0, 347, 0, 111, 0, 17, 0, 1, 0, 903, 0, 1061, 0, 541, 0, 143, 0

Offset: 0

Author

Clark Kimberling, Sep 21 2014

Keywords

• nonn
• tabl
• easy

Examples

			First 9 rows:
1
0 ... 1
1 ... 0 ... 1
0 ... 3 ... 0 ... 1
3 ... 0 ... 5 ... 0 ... 1
0 ... 11 .. 0 ... 7 ... 0 ...1
11 .. 0 ... 23 .. 0 ... 9 ... 0 ... 1
0 ... 45 .. 0 ... 39 .. 0 ... 11 .. 0 ... 1
45 .. 0 ... 107 . 0 ... 59 .. 0 ... 13 .. 0 ... 1
T(3,1) counts these 3 paths given as vector sums applied to (0,0):
(1,1) + (1,-1), (2,0), (1,-1) + (1,1).
		

Links

• Clark Kimberling, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A247623, A247629, A026300, A001003 (1st column of this triangle).

Programs

Mathematica

t[0, 0] = 1; t[1, 1] = 1; t[2, 0] = 1; t[2, 2] = 1; t[n_, k_] := t[n, k] = If[n >= 2 && k >= 1, t[n - 1, k - 1] + t[n - 1, k + 1] + t[n - 2, k], 0]; t[n_, 0] := t[n, 0] = t[n - 1, 1]; u = Table[t[n, k], {n, 0, 16}, {k, 0, n}];
v = Flatten[u] (* A247622 sequence *)
TableForm[u]   (* A247622 array *)
Map[Total, u]  (* A247623 *)