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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.
10
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
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, 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
KEYWORD
nonn,tabf,walk
AUTHOR
Alois P. Heinz, Mar 26 2017
STATUS
approved