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%I #19 Jul 22 2020 16:49:24
%S 1,0,0,0,0,3,0,0,0,6,3,0,0,14,12,1,0,36,39,6,0,98,120,25,0,276,363,90,
%T 0,794,1092,301,0,2316,3279,966,0,6818,9840,3025,0,20196,29523,9330,0,
%U 60074,88572,28501,0,179196,265719,86526
%N Rectangular array read by rows: T(n,k) is the number of words of length n on alphabet {0,1,2} that have exactly k records, n>=0, 0<=k<=3.
%C A record in a word a_1,a_2,...,a_n is a letter a_j that is larger than all the preceding letters. That is, a_j>a_i for all i<j.
%H Philippe Flajolet and Robert Sedgewick, <a href="http://algo.inria.fr/flajolet/Publications/AnaCombi/anacombi.html">Analytic Combinatorics</a>, Cambridge Univ. Press, 2009.
%F G.f.: Product_{j=1..3} (1 + y*x/(1 - j*x)). Generally for words on alphabet {0,1,...,r} the o.g.f. is Product_{j=1..r} (1 + y*x/(1 - j*x)).
%e 1, 0, 0, 0;
%e 0, 3, 0, 0;
%e 0, 6, 3, 0;
%e 0, 14, 12, 1;
%e 0, 36, 39, 6;
%e 0, 98, 120, 25;
%e 0, 276, 363, 90;
%e 0, 794, 1092, 301;
%e 0, 2316, 3279, 966;
%t nn = 12;CoefficientList[Series[Product[1 + u z/(1 - j z), {j, 1, 3}], {z, 0, nn}], {z,u}] // Grid
%Y Column k=0 gives A000007.
%Y Column k=1 gives A001550.
%Y Column k=2 gives A029858.
%Y Column k=3 gives A000392.
%Y Row sums give A000244.
%K nonn,tabf
%O 0,6
%A _Geoffrey Critzer_, Apr 27 2017
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