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%I #27 Dec 08 2017 15:12:34
%S 2,3,3,3,4,4,3,4,4,3,4,4,3,4,5,5,3,4,5,5,3,4,4,3,4,5,5,3,4,5,5,3,4,4,
%T 3,4,5,5,3,4,5,5,3,4,4,3,4,5,5,3,4,5,6,6,3,4,5,6,6,3,4,4,3,4,5,5,3,4,
%U 5,6,6,3,4,5,6,6,3,4,4,3,4,5,5,3,4,5,5,3,4,4,3,4,5,5,3,4,5,6,6,3,4,5,6,6,3,4
%N Irregular triangle, read by rows: the Schroeder generating tree, read from left to right, row by row, starting at the root.
%C Row n has A006318(n-1) terms (these are the large Schroeder numbers).
%C The limiting sequence of the rows is A295570.
%H Rémy Sigrist, <a href="/A295569/b295569.txt">Rows n = 1..9 of triangle, flattened</a>
%H D. Kremer, <a href="https://doi.org/10.1016/S0012-365X(99)00302-7">Permutations with forbidden subsequences and a generalized Schröder number</a>, Discrete Math. 218 (2000) 121-130.
%H Julian West, <a href="http://dx.doi.org/10.1016/0012-365X(94)00067-1">Generating trees and the Catalan and Schröder numbers</a>, Discrete Math. 146 (1995), 247-262.
%H Julian West, <a href="https://doi.org/10.1016/S0012-365X(96)83023-8">Generating trees and forbidden subsequences</a>, Discrete Math., 157 (1996), 363-374.
%e The triangle starts with a root node (at level 1) labeled 2; thereafter every node labeled k has k children at the next level whose labels are 3, 4, ..., k, k+1, k+1.
%e Rows 1, 2, 3, 4, and part of 5 are:
%e 2,
%e 3,3,
%e 3,4,4,3,4,4,
%e 3,4,4,3,4,5,5,3,4,5,5,3,4,4,3,4,5,5,3,4,5,5,
%e 3,4,4,3,4,5,5,3,4,5,5,3,4,4,3,4,5,5,3,4,5,6,6,3,4,5,6,6,...
%e ...
%p with(ListTools);
%p psi:=proc(S)
%p Flatten(subs( {2=[3,3], 3=[3,4,4], 4=[3,4,5,5], 5=[3,4,5,6,6], 6=[3,4,5,6,7,7], 7=[3,4,5,6,7,8,8]}, S)); # This will only work for the first 7 generations. For further generations, extend the "subs" command
%p end;
%p S:=[2];
%p for n from 1 to 6 do S:=psi(S): od:
%p S;
%Y Cf. A006318, A295568, A295570.
%K nonn,tabf
%O 1,1
%A _N. J. A. Sloane_, Nov 29 2017
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