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Irregular triangle read by rows: T(n,k) is the total number of unmatched parentheses (both left and right) in the k-th string of parentheses of length n, where strings within a row are in reverse lexicographical order.
4

%I #24 Mar 13 2024 04:42:39

%S 0,1,1,2,2,0,2,3,3,1,3,1,1,1,3,4,4,2,4,2,2,2,4,2,2,0,2,0,2,2,4,5,5,3,

%T 5,3,3,3,5,3,3,1,3,1,3,3,5,3,3,1,3,1,1,1,3,1,1,1,3,1,3,3,5,6,6,4,6,4,

%U 4,4,6,4,4,2,4,2,4,4,6,4,4,2,4,2,2,2,4,2

%N Irregular triangle read by rows: T(n,k) is the total number of unmatched parentheses (both left and right) in the k-th string of parentheses of length n, where strings within a row are in reverse lexicographical order.

%C See A370883 for more information.

%D Donald E. Knuth, The Art of Computer Programming, Vol. 4A: Combinatorial Algorithms, Part 1, Addison-Wesley, 2011, Section 7.2.1.6, p. 459.

%H Paolo Xausa, <a href="/A370885/b370885.txt">Table of n, a(n) for n = 0..16382</a> (rows 0..13 of the triangle, flattened).

%F T(n,k) = A370883(n,k) + A370884(n,k).

%e Triangle begins:

%e [0] 0;

%e [1] 1 1;

%e [2] 2 2 0 2;

%e [3] 3 3 1 3 1 1 1 3;

%e [4] 4 4 2 4 2 2 2 4 2 2 0 2 0 2 2 4;

%e ...

%t countLR[s_] := StringLength[s] - StringLength[StringJoin[StringCases[s, RegularExpression["1(?R)*+0"]]]];

%t Array[Map[countLR, IntegerString[Range[0, 2^#-1], 2, #]] &, 7, 0]

%Y Cf. A370883, A370884.

%Y Cf. A000079 (row lengths).

%Y Apparently, row sums are given by 2*A189391.

%K nonn,tabf

%O 0,4

%A _Paolo Xausa_, Mar 06 2024