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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.

%I A284652 #21 Oct 09 2019 13:35:22
%S A284652 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,
%T A284652 323,1,15,87,269,565,370,835,10,116,370,896,1766,983,2188,9,99,499,
%U A284652 1473,2776,5446,2627,5798,4,91,536,2290,5528,8657,16655,7086,15511
%N 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.
%H A284652 Alois P. Heinz, <a href="/A284652/b284652.txt">Columns k = 0..160, flattened</a>
%H A284652 Wikipedia, <a href="https://en.wikipedia.org/wiki/Lattice_path">Lattice path</a>
%H A284652 Wikipedia, <a href="https://en.wikipedia.org/wiki/Self-avoiding_walk">Self-avoiding walk</a>
%F A284652 Sum_{k=n..n*(n+3)/2} (k+1) * T(n,k) = A284231(n).
%e A284652 Triangle T(n,k) begins:
%e A284652 1;
%e A284652 .  1, 1;
%e A284652 .  .  2, 1, 1,  1;
%e A284652 .  .  .  4, 4,  4,  7,   3,   1,   1;
%e A284652 .  .  .  .  9,  8, 16,  21,  17,  15,   10,    9, ... ;
%e A284652 .  .  .  .  .  21, 22,  54,  87,  87,  116,   99, ... ;
%e A284652 .  .  .  .  .   .  51,  54, 178, 269,  370,  499, ... ;
%e A284652 .  .  .  .  .   .   .  127, 142, 565,  896, 1473, ... ;
%e A284652 .  .  .  .  .   .   .    .  323, 370, 1766, 2776, ... ;
%e A284652 .  .  .  .  .   .   .    .    .  835,  983, 5446, ... ;
%e A284652 .  .  .  .  .   .   .    .    .     . 2188, 2627, ... ;
%Y A284652 Row sums give A284230.
%Y A284652 Column sums give A284415.
%Y A284652 Antidiagonal sums give A284428.
%Y A284652 T(n,n) gives A001006.
%Y A284652 T(n,n+1) gives A284778.
%Y A284652 T(n,2n) gives A284416.
%Y A284652 T(n,n*(n+1)/2) gives A284418.
%Y A284652 Column heights give A122797(k+1).
%Y A284652 Cf. A000096, A284231, A284461, A284414 (this triangle read by rows).
%K A284652 nonn,tabf,walk
%O A284652 0,4
%A A284652 _Alois P. Heinz_, Mar 31 2017