A284414 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), n>=0, n<=k<=n*(n+3)/2, read by rows.

1, 1, 1, 2, 1, 1, 1, 4, 4, 4, 7, 3, 1, 1, 9, 8, 16, 21, 17, 15, 10, 9, 4, 1, 1, 21, 22, 54, 87, 87, 116, 99, 91, 78, 42, 31, 17, 10, 5, 1, 1, 51, 54, 178, 269, 370, 499, 536, 590, 560, 510, 420, 350, 268, 185, 132, 69, 44, 23, 11, 6, 1, 1

Offset

0,4

Links

Alois P. Heinz, Rows n = 0..50, flattened

Alois P. Heinz, Animation of T(5,12)=91 walks

Wikipedia, Lattice_path

Wikipedia, Self-avoiding walk

Formula

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

Example

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,  4,  1,  1;
.  .  .  .  .  21, 22, 54, 87, 87, 116, 99, 91, 78, 42, 31, 17, 10, 5, 1, 1;

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.

Cf. A000096, A284231, A284461, A284652 (this triangle read by columns).

Sequence in context: A213268 A291118 A245184 * A355614 A140274 A095231

Adjacent sequences: A284411 A284412 A284413 * A284415 A284416 A284417

Keyword

nonn,tabf,walk

Author

Alois P. Heinz, Mar 26 2017

Status

approved