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A370885
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
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, 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, 4, 4, 6, 4, 4, 2, 4, 2, 4, 4, 6, 4, 4, 2, 4, 2, 2, 2, 4, 2
OFFSET
0,4
COMMENTS
See A370883 for more information.
REFERENCES
Donald E. Knuth, The Art of Computer Programming, Vol. 4A: Combinatorial Algorithms, Part 1, Addison-Wesley, 2011, Section 7.2.1.6, p. 459.
LINKS
Paolo Xausa, Table of n, a(n) for n = 0..16382 (rows 0..13 of the triangle, flattened).
FORMULA
T(n,k) = A370883(n,k) + A370884(n,k).
EXAMPLE
Triangle begins:
[0] 0;
[1] 1 1;
[2] 2 2 0 2;
[3] 3 3 1 3 1 1 1 3;
[4] 4 4 2 4 2 2 2 4 2 2 0 2 0 2 2 4;
...
MATHEMATICA
countLR[s_] := StringLength[s] - StringLength[StringJoin[StringCases[s, RegularExpression["1(?R)*+0"]]]];
Array[Map[countLR, IntegerString[Range[0, 2^#-1], 2, #]] &, 7, 0]
CROSSREFS
Cf. A000079 (row lengths).
Apparently, row sums are given by 2*A189391.
Sequence in context: A256750 A228430 A241533 * A072738 A165316 A362991
KEYWORD
nonn,tabf
AUTHOR
Paolo Xausa, Mar 06 2024
STATUS
approved