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%I #18 Oct 26 2018 20:37:12
%S 1,1,1,1,1,0,1,1,1,0,1,1,2,2,0,1,1,3,6,5,0,1,1,4,12,21,16,0,1,1,5,20,
%T 54,87,61,0,1,1,6,30,110,276,413,271,0,1,1,7,42,195,670,1574,2213,
%U 1372,0,1,1,8,56,315,1380,4470,9916,13205,7795,0
%N Number A(n,k) of k-ascent sequences of length n with no consecutive repeated letters; square array A(n,k), n>=0, k>=0, read by antidiagonals.
%H Alois P. Heinz, <a href="/A264909/b264909.txt">Antidiagonals n = 0..140, flattened</a>
%H S. Kitaev, J. Remmel, <a href="https://arxiv.org/abs/1503.00914">p-Ascent Sequences</a>, arXiv:1503.00914 [math.CO], 2015.
%e Square array A(n,k) begins:
%e 1, 1, 1, 1, 1, 1, 1, 1, ...
%e 1, 1, 1, 1, 1, 1, 1, 1, ...
%e 0, 1, 2, 3, 4, 5, 6, 7, ...
%e 0, 2, 6, 12, 20, 30, 42, 56, ...
%e 0, 5, 21, 54, 110, 195, 315, 476, ...
%e 0, 16, 87, 276, 670, 1380, 2541, 4312, ...
%e 0, 61, 413, 1574, 4470, 10555, 21931, 41468, ...
%e 0, 271, 2213, 9916, 32440, 86815, 201761, 422128, ...
%p b:= proc(n, k, i, t) option remember; `if`(n<1, 1, add(
%p `if`(j=i, 0, b(n-1, k, j, t+`if`(j>i, 1, 0))), j=0..t+k))
%p end:
%p A:= (n, k)-> b(n-1, k, 0$2):
%p seq(seq(A(n, d-n), n=0..d), d=0..12);
%t b[n_, k_, i_, t_] := b[n, k, i, t] = If[n<1, 1, Sum[If[j == i, 0, b[n-1, k, j, t + If[j>i, 1, 0]]], {j, 0, t+k}]]; A[n_, k_] := b[n-1, k, 0, 0]; Table[Table[A[n, d-n], {n, 0, d}], {d, 0, 12}] // Flatten (* _Jean-François Alcover_, Feb 17 2016, after _Alois P. Heinz_ *)
%Y Columns k=1-10 give: A138265, A263852, A263853, A263854, A264910, A264911, A264912, A264913, A264914, A264915.
%Y Rows k=0+1,2-4 give: A000012, A001477, A002378, A160378(n+1).
%Y Main diagonal gives A264916.
%K nonn,tabl
%O 0,13
%A _Alois P. Heinz_, Nov 28 2015
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