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.

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. A370883, A370884.

Cf. A000079 (row lengths).

Apparently, row sums are given by 2*A189391.

Sequence in context: A256750 A228430 A241533 * A072738 A165316 A362991

Adjacent sequences: A370882 A370883 A370884 * A370886 A370887 A370888

Keyword

nonn,tabf

Author

Paolo Xausa, Mar 06 2024

Status

approved