login
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.
3
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
0,4
LINKS
Wikipedia, Lattice path
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, ... ;
. . . . . 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, ... ;
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).
Sequence in context: A007104 A102627 A296560 * A261242 A088296 A282738
KEYWORD
nonn,tabf,walk
AUTHOR
Alois P. Heinz, Mar 31 2017
STATUS
approved