A247629 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.

1, 0, 1, 1, 0, 1, 0, 3, 0, 1, 4, 0, 5, 0, 1, 0, 12, 0, 7, 0, 1, 16, 0, 24, 0, 9, 0, 1, 0, 52, 0, 40, 0, 11, 0, 1, 68, 0, 116, 0, 60, 0, 13, 0, 1, 0, 236, 0, 216, 0, 84, 0, 15, 0, 1, 304, 0, 568, 0, 360, 0, 112, 0, 17, 0, 1, 0, 1108, 0, 1144, 0, 556, 0, 144

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
4 ... 0 ... 5 ... 0 ... 1
0 ... 12 .. 0 ... 7 ... 0 ...1
16 .. 0 ... 24 .. 0 ... 9 ... 0 ... 1
0 ... 52 .. 0 ... 40 .. 0 ... 11 .. 0 ... 1
68 .. 0 ... 116 . 0 ... 60 .. 0 ... 13 .. 0 ... 1
T(4,2) counts these 5 paths given as vector sums applied to (0,0):
(1,1) + (1,1) + (1,1) + (1,-1)
(1,1) + (1,1) + (2,0)
(1,1) + (1,1) + (1,-1) + (1,1)
(1,1) + (2,0) + (1,1)
(1,1) + (1,-1) + (1,1) + (1,-1)
		

Links

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

Crossrefs

Cf. A247623, A247629, A026300, A006319 (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] = If[n >= 2, t[n - 2, 0] + t[n - 1, 1], 0]; u = Table[t[n, k], {n, 0, 16}, {k, 0, n}]; TableForm[u] (* A247629 array *)
v = Flatten[u] (* A247629 sequence *)
Map[Total, u] (* A247630 *)