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%I #18 Oct 18 2018 16:57:11
%S 1,1,0,1,1,0,1,1,1,0,1,1,6,1,0,1,1,6,43,1,0,1,1,6,90,352,1,0,1,1,6,90,
%T 1879,3114,1,0,1,1,6,90,2520,47024,29004,1,0,1,1,6,90,2520,102011,
%U 1331664,280221,1,0,1,1,6,90,2520,113400,5176504,41250519,2782476,1,0
%N Number A(n,k) of words on {1,1,2,2,...,n,n} with longest increasing subsequence of length <= k; square array A(n,k), n>=0, k>=0, read by antidiagonals.
%H Alois P. Heinz, <a href="/A267479/b267479.txt">Antidiagonals n = 0..30, flattened</a>
%H Ferenc Balogh, <a href="https://arxiv.org/abs/1505.01389">A generalization of Gessel's generating function to enumerate words with double or triple occurrences in each letter and without increasing subsequences of a given length</a>, arXiv:1505.01389, 2015
%H Shalosh B. Ekhad and Doron Zeilberger, <a href="http://www.math.rutgers.edu/~zeilberg/mamarim/mamarimhtml/sloane75.html">The Generating Functions Enumerating 12..d-Avoiding Words with r occurrences of each of 1,2, ..., n are D-finite for all d and all r</a>, 2014
%F A(n,k) = Sum_{i=0..k} A267480(n,i).
%e Square array A(n,k) begins:
%e 1, 1, 1, 1, 1, 1, 1, ...
%e 0, 1, 1, 1, 1, 1, 1, ...
%e 0, 1, 6, 6, 6, 6, 6, ...
%e 0, 1, 43, 90, 90, 90, 90, ...
%e 0, 1, 352, 1879, 2520, 2520, 2520, ...
%e 0, 1, 3114, 47024, 102011, 113400, 113400, ...
%e 0, 1, 29004, 1331664, 5176504, 7235651, 7484400, ...
%Y Columns k=0-4 give: A000007, A000012, A220097, A266734, A266735.
%Y Main diagonal gives A000680.
%Y First lower diagonal gives A267532.
%Y Cf. A214015, A267480.
%K nonn,tabl
%O 0,13
%A _Alois P. Heinz_, Jan 15 2016