A295569 Irregular triangle, read by rows: the Schroeder generating tree, read from left to right, row by row, starting at the root.

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, 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, 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

Offset

1,1

Comments

Row n has A006318(n-1) terms (these are the large Schroeder numbers).

The limiting sequence of the rows is A295570.

Links

Rémy Sigrist, Rows n = 1..9 of triangle, flattened

D. Kremer, Permutations with forbidden subsequences and a generalized Schröder number, Discrete Math. 218 (2000) 121-130.

Julian West, Generating trees and the Catalan and Schröder numbers, Discrete Math. 146 (1995), 247-262.

Julian West, Generating trees and forbidden subsequences, Discrete Math., 157 (1996), 363-374.

Example

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.
Rows 1, 2, 3, 4, and part of 5 are:
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,
3,4,4,3,4,5,5,3,4,5,5,3,4,4,3,4,5,5,3,4,5,6,6,3,4,5,6,6,...
...

Maple

with(ListTools);
psi:=proc(S)
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
end;
S:=[2];
for n from 1 to 6 do S:=psi(S): od:
S;

Crossrefs

Cf. A006318, A295568, A295570.

Sequence in context: A322974 A326201 A342625 * A129574 A323466 A130193

Adjacent sequences: A295566 A295567 A295568 * A295570 A295571 A295572

Keyword

nonn,tabf

Author

N. J. A. Sloane, Nov 29 2017

Status

approved