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A264909 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.

%I A264909 #18 Oct 26 2018 20:37:12
%S A264909 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 A264909 54,87,61,0,1,1,6,30,110,276,413,271,0,1,1,7,42,195,670,1574,2213,
%U A264909 1372,0,1,1,8,56,315,1380,4470,9916,13205,7795,0
%N A264909 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 A264909 Alois P. Heinz, <a href="/A264909/b264909.txt">Antidiagonals n = 0..140, flattened</a>
%H A264909 S. Kitaev, J. Remmel, <a href="https://arxiv.org/abs/1503.00914">p-Ascent Sequences</a>, arXiv:1503.00914 [math.CO], 2015.
%e A264909 Square array A(n,k) begins:
%e A264909   1,   1,    1,    1,     1,     1,      1,      1, ...
%e A264909   1,   1,    1,    1,     1,     1,      1,      1, ...
%e A264909   0,   1,    2,    3,     4,     5,      6,      7, ...
%e A264909   0,   2,    6,   12,    20,    30,     42,     56, ...
%e A264909   0,   5,   21,   54,   110,   195,    315,    476, ...
%e A264909   0,  16,   87,  276,   670,  1380,   2541,   4312, ...
%e A264909   0,  61,  413, 1574,  4470, 10555,  21931,  41468, ...
%e A264909   0, 271, 2213, 9916, 32440, 86815, 201761, 422128, ...
%p A264909 b:= proc(n, k, i, t) option remember; `if`(n<1, 1, add(
%p A264909       `if`(j=i, 0, b(n-1, k, j, t+`if`(j>i, 1, 0))), j=0..t+k))
%p A264909     end:
%p A264909 A:= (n, k)-> b(n-1, k, 0$2):
%p A264909 seq(seq(A(n, d-n), n=0..d), d=0..12);
%t A264909 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 A264909 Columns k=1-10 give: A138265, A263852, A263853, A263854, A264910, A264911, A264912, A264913, A264914, A264915.
%Y A264909 Rows k=0+1,2-4 give: A000012, A001477, A002378, A160378(n+1).
%Y A264909 Main diagonal gives A264916.
%K A264909 nonn,tabl
%O A264909 0,13
%A A264909 _Alois P. Heinz_, Nov 28 2015