%I #26 Mar 13 2024 04:42:32

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

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

%U 1,1,3,0,1,0,2,0,2,2,4,0,1,0,2,0,1,1,3,0

%N Irregular triangle read by rows: T(n,k) is the number of unmatched left parentheses 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.

%C The first half of each row n >= 1 is equal to row n-1.

%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="/A370884/b370884.txt">Table of n, a(n) for n = 0..16382</a> (rows 0..13 of the triangle, flattened).

%F T(n,k) = A370885(n,k) - A370883(n,k).

%e Triangle begins:

%e [0] 0;

%e [1] 0 1;

%e [2] 0 1 0 2;

%e [3] 0 1 0 2 0 1 1 3;

%e [4] 0 1 0 2 0 1 1 3 0 1 0 2 0 2 2 4;

%e ...

%e The strings corresponding to row 2, in reverse lexicographical order, are:

%e "))" (0 unmatched left parentheses),

%e ")(" (1 unmatched left parenthesis),

%e "()" (0 unmatched left parentheses), and

%e "((" (2 unmatched left parentheses).

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

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

%Y Cf. A370883, A370885.

%Y Cf. A000079 (row lengths).

%Y Apparently, row sums are given by A189391.

%K nonn,tabf

%O 0,7

%A _Paolo Xausa_, Mar 06 2024