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%I #27 Apr 02 2017 19:49:44
%S 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,
%T 87,116,99,91,78,42,31,17,10,5,1,1,51,54,178,269,370,499,536,590,560,
%U 510,420,350,268,185,132,69,44,23,11,6,1,1
%N 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.
%H Alois P. Heinz, <a href="/A284414/b284414.txt">Rows n = 0..50, flattened</a>
%H Alois P. Heinz, <a href="/A284414/a284414.gif">Animation of T(5,12)=91 walks</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Lattice_path">Lattice_path</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Self-avoiding_walk">Self-avoiding walk</a>
%F Sum_{k=n..n*(n+3)/2} (k+1) * T(n,k) = A284231(n).
%e Triangle T(n,k) begins:
%e 1;
%e . 1, 1;
%e . . 2, 1, 1, 1;
%e . . . 4, 4, 4, 7, 3, 1, 1;
%e . . . . 9, 8, 16, 21, 17, 15, 10, 9, 4, 1, 1;
%e . . . . . 21, 22, 54, 87, 87, 116, 99, 91, 78, 42, 31, 17, 10, 5, 1, 1;
%Y Row sums give A284230.
%Y Column sums give A284415.
%Y Antidiagonal sums give A284428.
%Y T(n,n) gives A001006.
%Y T(n,n+1) gives A284778.
%Y T(n,2n) gives A284416.
%Y T(n,n*(n+1)/2) gives A284418.
%Y Cf. A000096, A284231, A284461, A284652 (this triangle read by columns).
%K nonn,tabf,walk
%O 0,4
%A _Alois P. Heinz_, Mar 26 2017
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