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A267480 Number T(n,k) of words on {1,1,2,2,...,n,n} with longest increasing subsequence of length k; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

%I A267480 #19 May 09 2018 09:51:00
%S A267480 1,0,1,0,1,5,0,1,42,47,0,1,351,1527,641,0,1,3113,43910,54987,11389,0,
%T A267480 1,29003,1302660,3844840,2059147,248749,0,1,280220,40970298,265777225,
%U A267480 285588543,82025038,6439075,0,1,2782475,1364750889,19104601915,37783672691,19773928713,3507289363,192621953
%N A267480 Number T(n,k) of words on {1,1,2,2,...,n,n} with longest increasing subsequence of length k; triangle T(n,k), n>=0, 0<=k<=n, read by rows.
%H A267480 Alois P. Heinz, <a href="/A267480/b267480.txt">Rows n = 0..18, flattened</a>
%H A267480 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 A267480 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 A267480 T(n,0) = A267479(n,0), T(n,k) = A267479(n,k) - A267479(n,k-1) for k>0.
%F A267480 Sum_{k=0..n-1} T(n,k) = A267532(n).
%e A267480 Triangle T(n,k) begins:
%e A267480   1;
%e A267480   0, 1;
%e A267480   0, 1,      5;
%e A267480   0, 1,     42,       47;
%e A267480   0, 1,    351,     1527,       641;
%e A267480   0, 1,   3113,    43910,     54987,     11389;
%e A267480   0, 1,  29003,  1302660,   3844840,   2059147,   248749;
%e A267480   0, 1, 280220, 40970298, 265777225, 285588543, 82025038, 6439075;
%Y A267480 Main diagonal gives A006902.
%Y A267480 Row sums give A000680.
%Y A267480 Cf. A267479, A267532.
%K A267480 nonn,tabl
%O A267480 0,6
%A A267480 _Alois P. Heinz_, Jan 15 2016