A284652 Number T(n,k) of self-avoiding planar walks of length k starting at (0,0), ending at (n,0), remaining in the first quadrant and using steps (0,1), (1,0), (1,1), (-1,1), and (1,-1) with the restriction that (0,1) is never used below the diagonal and (1,0) is never used above the diagonal; triangle T(n,k), k>=0, floor((sqrt(1+8*k)-1)/2)<=n<=k, read by columns.

1, 1, 1, 2, 1, 4, 1, 4, 9, 1, 4, 8, 21, 7, 16, 22, 51, 3, 21, 54, 54, 127, 1, 17, 87, 178, 142, 323, 1, 15, 87, 269, 565, 370, 835, 10, 116, 370, 896, 1766, 983, 2188, 9, 99, 499, 1473, 2776, 5446, 2627, 5798, 4, 91, 536, 2290, 5528, 8657, 16655, 7086, 15511

Offset: 0

Author

Alois P. Heinz, Mar 31 2017

Keywords

• nonn
• tabf
• walk

Examples

			Triangle T(n,k) begins:
1;
.  1, 1;
.  .  2, 1, 1,  1;
.  .  .  4, 4,  4,  7,   3,   1,   1;
.  .  .  .  9,  8, 16,  21,  17,  15,   10,    9, ... ;
.  .  .  .  .  21, 22,  54,  87,  87,  116,   99, ... ;
.  .  .  .  .   .  51,  54, 178, 269,  370,  499, ... ;
.  .  .  .  .   .   .  127, 142, 565,  896, 1473, ... ;
.  .  .  .  .   .   .    .  323, 370, 1766, 2776, ... ;
.  .  .  .  .   .   .    .    .  835,  983, 5446, ... ;
.  .  .  .  .   .   .    .    .     . 2188, 2627, ... ;
		

Links

• Alois P. Heinz, Columns k = 0..160, flattened
• Wikipedia, Lattice path
• Wikipedia, Self-avoiding walk

Crossrefs

Row sums give A284230.

Column sums give A284415.

Antidiagonal sums give A284428.

T(n,n) gives A001006.

T(n,n+1) gives A284778.

T(n,2n) gives A284416.

T(n,n*(n+1)/2) gives A284418.

Column heights give A122797(k+1).

Cf. A000096, A284231, A284461, A284414 (this triangle read by rows).

Formula

Sum_{k=n..n*(n+3)/2} (k+1) * T(n,k) = A284231(n).