A247311 Rectangular array read upwards by columns: T = T(n,k) = number of paths from (0,1) to (n,k), where 0 <= k <= 2, consisting of segments given by the vectors (1,1), (1,0), (1,-1).

1, 0, 0, 1, 1, 0, 2, 2, 1, 4, 5, 3, 9, 12, 8, 21, 29, 20, 50, 70, 49, 120, 169, 119, 289, 408, 288, 697, 985, 696, 1682, 2378, 1681, 4060, 5741, 4059, 9801, 13860, 9800, 23661, 33461, 23660, 57122, 80782, 57121, 137904, 195025, 137903, 332929, 470832, 332928

Offset: 0

Author

Clark Kimberling, Sep 12 2014

Keywords

• nonn
• tabf
• easy

Comments

Also, T(n,k) = number of strings s(0)..s(n) of integers such that s(0) = 0, s(n) = k, and for 0 < i <= n, s(i) is in {0,1,2}, and s(i) - s(i-1) is in {-1,0,1}.

(row 0, the bottom row): A024537;

(row 1, the middle row): A000129;

(row 2, the top row): A048739;

(n-th column sum): A000129.

Examples

			First 10 columns:
  0 .. 0 .. 1 .. 3 .. 8 ... 20 .. 49 .. 119 .. 288 .. 696
  0 .. 1 .. 2 .. 5 .. 12 .. 29 .. 70 .. 169 .. 408 .. 985
  1 .. 1 .. 2 .. 4 .. 9 ... 21 .. 50 .. 120 .. 289 .. 697
T(3,2) counts these 3 paths, given as vector sums applied to (0,0):
  (1,1) + (1,1) + (1,0); (1,1) + (1,0) + (1,1); (1,0) + (1,1) + (1,1).
		

Links

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

Crossrefs

Cf. A247049, A247309, A247310, A000129, A024537, A048739.

Programs

Mathematica

t[0, 0] = 1; t[0, 1] = 0; t[0, 2] = 0; t[1, 2] = 0;
t[n_, 0] := t[n, 0] = t[n - 1, 0] + t[n - 1, 1];
t[n_, 1] := t[n, 1] = t[n - 1, 0] + t[n - 1, 1] + t[n - 1, 2];
t[n_, 2] := t[n, 2] = t[n - 1, 1] + t[n - 1, 2]
TableForm[Reverse[Transpose[Table[t[n, k], {n, 0, 12}, {k, 0, 2}]]]] (*  array *)
Flatten[Table[t[n, k], {n, 0, 20}, {k, 0, 2}]] (* A247311 *)