A370220 -id:A370220 - cp's OEIS frontend

A370290 Row sums of A370220.

1, 4, 3, 9, 8, 8, 7, 6, 16, 15, 15, 14, 13, 15, 14, 14, 13, 12, 13, 12, 11, 10, 25, 24, 24, 23, 22, 24, 23, 23, 22, 21, 22, 21, 20, 19, 24, 23, 23, 22, 21, 23, 22, 22, 21, 20, 21, 20, 19, 18, 22, 21, 21, 20, 19, 20, 19, 18, 17, 19, 18, 17, 16, 15, 36, 35, 35, 34

Offset: 1

Paolo Xausa, Feb 14 2024

nonn

See A370220 for more information.

Paolo Xausa, Table of n, a(n) for n = 1..10000

Cf. A370220.

zsums[m_] := With[{r = 2*Range[2, m]}, Reverse[Map[Total[#]+1 &, Select[Subsets[Range[2, 2*m-1], {m-1}], Min[r-#] > 0 &]]]];
Flatten[Array[zsums, 6]]

A370222 Irregular triangle T(n,k) read by rows: row n gives the inversion table (see comments) of the permutation encoded by row n of A370221.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 1, 1, 0, 1, 2, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 1, 2, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 1, 0, 0, 1, 1, 1, 0, 1, 1, 2, 0, 1, 2, 0, 0, 1, 2, 1, 0, 1, 2, 2, 0, 1, 2, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0

Offset: 1

Paolo Xausa, Feb 12 2024

Knuth (2011) uses the inversion table c_1, c_2, ..., c_k (defined so that exactly c_k elements to the right of k are less than k) to encode the permutation given by row n of A370221.
This way c_1 = 0 and 0 <= c_(k+1) <= c_k + 1 for 1 <= k < m, where m >= 1 is half the length of the corresponding properly nested string of parentheses (see example).
The concatenation of terms in each row from row n = 23714 to 82498 (corresponding to strings of length 22, excluding the last one) gives the A000108(11) - 1 = 58785 terms of the finite sequence A239903.

			The following table lists c_k values for properly nested strings having lengths up to 8, along with d_k, z_k and p_k values from related combinatorial objects (see related sequences for more information). Cf. Knuth (2011), p. 442, Table 1.
.
      | Properly |          | A370219 | A370220 | A370221 |
      | Nested   | A063171  | d d d d | z z z z | p p p p | c c c c
    n | String   |   (n)    | 1 2 3 4 | 1 2 3 4 | 1 2 3 4 | 1 2 3 4
  ----+----------+----------+---------+---------+---------+---------
    1 | ()       | 10       | 1       | 1       | 1       | 0
    2 | ()()     | 1010     | 1 1     | 1 3     | 1 2     | 0 0
    3 | (())     | 1100     | 0 2     | 1 2     | 2 1     | 0 1
    4 | ()()()   | 101010   | 1 1 1   | 1 3 5   | 1 2 3   | 0 0 0
    5 | ()(())   | 101100   | 1 0 2   | 1 3 4   | 1 3 2   | 0 0 1
    6 | (())()   | 110010   | 0 2 1   | 1 2 5   | 2 1 3   | 0 1 0
    7 | (()())   | 110100   | 0 1 2   | 1 2 4   | 2 3 1   | 0 1 1
    8 | ((()))   | 111000   | 0 0 3   | 1 2 3   | 3 2 1   | 0 1 2
    9 | ()()()() | 10101010 | 1 1 1 1 | 1 3 5 7 | 1 2 3 4 | 0 0 0 0
   10 | ()()(()) | 10101100 | 1 1 0 2 | 1 3 5 6 | 1 2 4 3 | 0 0 0 1
   11 | ()(())() | 10110010 | 1 0 2 1 | 1 3 4 7 | 1 3 2 4 | 0 0 1 0
   12 | ()(()()) | 10110100 | 1 0 1 2 | 1 3 4 6 | 1 3 4 2 | 0 0 1 1
   13 | ()((())) | 10111000 | 1 0 0 3 | 1 3 4 5 | 1 4 3 2 | 0 0 1 2
   14 | (())()() | 11001010 | 0 2 1 1 | 1 2 5 7 | 2 1 3 4 | 0 1 0 0
   15 | (())(()) | 11001100 | 0 2 0 2 | 1 2 5 6 | 2 1 4 3 | 0 1 0 1
   16 | (()())() | 11010010 | 0 1 2 1 | 1 2 4 7 | 2 3 1 4 | 0 1 1 0
   17 | (()()()) | 11010100 | 0 1 1 2 | 1 2 4 6 | 2 3 4 1 | 0 1 1 1
   18 | (()(())) | 11011000 | 0 1 0 3 | 1 2 4 5 | 2 4 3 1 | 0 1 1 2
   19 | ((()))() | 11100010 | 0 0 3 1 | 1 2 3 7 | 3 2 1 4 | 0 1 2 0
   20 | ((())()) | 11100100 | 0 0 2 2 | 1 2 3 6 | 3 2 4 1 | 0 1 2 1
   21 | ((()())) | 11101000 | 0 0 1 3 | 1 2 3 5 | 3 4 2 1 | 0 1 2 2
   22 | (((()))) | 11110000 | 0 0 0 4 | 1 2 3 4 | 4 3 2 1 | 0 1 2 3

Donald E. Knuth, The Art of Computer Programming, Vol. 4A: Combinatorial Algorithms, Part 1, Addison-Wesley, 2011, Section 7.2.1.6, pp. 440-444.

Paolo Xausa, Table of n, a(n) for n = 1..15521 (rows 1..2055 of the triangle, flattened).

Cf. A000108, A063171, A072643 (row lengths), A239903.

Cf. A370219, A370220, A370221, A370292 (row sums).

clist[m_] := With[{r = 2*Range[2, m]-1}, Reverse[Map[Join[{0}, r-#] &, Select[Subsets[Range[2, 2*m-1], {m-1}], Min[r-#] >= 0 &]]]];
Array[Delete[clist[#], 0] &, 5]
(* 2nd program: uses Algorithm Z from Knuth's TAOCP section 7.2.1.6, exercise 2 *)
zlist[m_] := Block[{z = 2*Range[m] - 1, j},
    Reap[
    While[True,
        Sow[z];
        If[z[[m-1]] < z[[m]] - 1,
            z[[m]]--,
            j = m - 1; z[[m]] = 2*m - 1;
            While[j > 1 && z[[j-1]] == z[[j]] - 1, z[[j]] = 2*j - 1; j--];
            If[j == 1,Break[]];
            z[[j]]--]
    ]][[2]][[1]]];
Join[{{0}}, Table[Delete[Map[2*Range[n] - 1 - # &, zlist[n]], 0], {n, 2, 5}]] (* Paolo Xausa, Mar 25 2024 *)

T(n,k) = 2*k - 1 - A370220(n,k).

A370221 Irregular triangle T(n,k) read by rows: row n lists the values encoding a permutation (see comments) related to the properly nested string of parentheses encoded by A063171(n).

Original entry on oeis.org

1, 1, 2, 2, 1, 1, 2, 3, 1, 3, 2, 2, 1, 3, 2, 3, 1, 3, 2, 1, 1, 2, 3, 4, 1, 2, 4, 3, 1, 3, 2, 4, 1, 3, 4, 2, 1, 4, 3, 2, 2, 1, 3, 4, 2, 1, 4, 3, 2, 3, 1, 4, 2, 3, 4, 1, 2, 4, 3, 1, 3, 2, 1, 4, 3, 2, 4, 1, 3, 4, 2, 1, 4, 3, 2, 1, 1, 2, 3, 4, 5, 1, 2, 3, 5, 4, 1, 2, 4, 3, 5

Offset: 1

Paolo Xausa, Feb 12 2024

Knuth (2011) refers to these terms as p_k and defines them so that, in a properly nested string of parentheses, the k-th right parenthesis matches the p_k-th left parenthesis.

A370222 gives the corresponding values of a related inversion table.

			The following table lists p_k values for properly nested strings having lengths up to 8, along with d_k, z_k and c_k values from related combinatorial objects (see related sequences for more information). Cf. Knuth (2011), p. 442, Table 1.
.
      | Properly |          | A370219 | A370220 |         | A370222
      | Nested   | A063171  | d d d d | z z z z | p p p p | c c c c
    n | String   |   (n)    | 1 2 3 4 | 1 2 3 4 | 1 2 3 4 | 1 2 3 4
  ----+----------+----------+---------+---------+---------+---------
    1 | ()       | 10       | 1       | 1       | 1       | 0
    2 | ()()     | 1010     | 1 1     | 1 3     | 1 2     | 0 0
    3 | (())     | 1100     | 0 2     | 1 2     | 2 1     | 0 1
    4 | ()()()   | 101010   | 1 1 1   | 1 3 5   | 1 2 3   | 0 0 0
    5 | ()(())   | 101100   | 1 0 2   | 1 3 4   | 1 3 2   | 0 0 1
    6 | (())()   | 110010   | 0 2 1   | 1 2 5   | 2 1 3   | 0 1 0
    7 | (()())   | 110100   | 0 1 2   | 1 2 4   | 2 3 1   | 0 1 1
    8 | ((()))   | 111000   | 0 0 3   | 1 2 3   | 3 2 1   | 0 1 2
    9 | ()()()() | 10101010 | 1 1 1 1 | 1 3 5 7 | 1 2 3 4 | 0 0 0 0
   10 | ()()(()) | 10101100 | 1 1 0 2 | 1 3 5 6 | 1 2 4 3 | 0 0 0 1
   11 | ()(())() | 10110010 | 1 0 2 1 | 1 3 4 7 | 1 3 2 4 | 0 0 1 0
   12 | ()(()()) | 10110100 | 1 0 1 2 | 1 3 4 6 | 1 3 4 2 | 0 0 1 1
   13 | ()((())) | 10111000 | 1 0 0 3 | 1 3 4 5 | 1 4 3 2 | 0 0 1 2
   14 | (())()() | 11001010 | 0 2 1 1 | 1 2 5 7 | 2 1 3 4 | 0 1 0 0
   15 | (())(()) | 11001100 | 0 2 0 2 | 1 2 5 6 | 2 1 4 3 | 0 1 0 1
   16 | (()())() | 11010010 | 0 1 2 1 | 1 2 4 7 | 2 3 1 4 | 0 1 1 0
   17 | (()()()) | 11010100 | 0 1 1 2 | 1 2 4 6 | 2 3 4 1 | 0 1 1 1
   18 | (()(())) | 11011000 | 0 1 0 3 | 1 2 4 5 | 2 4 3 1 | 0 1 1 2
   19 | ((()))() | 11100010 | 0 0 3 1 | 1 2 3 7 | 3 2 1 4 | 0 1 2 0
   20 | ((())()) | 11100100 | 0 0 2 2 | 1 2 3 6 | 3 2 4 1 | 0 1 2 1
   21 | ((()())) | 11101000 | 0 0 1 3 | 1 2 3 5 | 3 4 2 1 | 0 1 2 2
   22 | (((()))) | 11110000 | 0 0 0 4 | 1 2 3 4 | 4 3 2 1 | 0 1 2 3

Donald E. Knuth, The Art of Computer Programming, Vol. 4A: Combinatorial Algorithms, Part 1, Addison-Wesley, 2011, Section 7.2.1.6, pp. 440-444.

Paolo Xausa, Table of n, a(n) for n = 1..15521 (rows 1..2055 of the triangle, flattened).

Cf. A000108, A063171, A072643 (row lengths).

Cf. A370219, A370220, A370222, A370291 (row sums).

slist[m_] := Reverse[Select[Permutations[PadLeft[Table[-1, m], 2*m, 1]], Min[Accumulate[#]] >= 0 &]];
plist[s_] := Flatten[Reap[Module[{p, p0 = Flatten[Position[s, -1]], p1 = Flatten[Position[s, 1]], p1r}, p1r = p1; For[i = 1, i <= Length[p0], i++, p = Max[Select[p1r, # < p0[[i]] &]]; Sow[Position[p1, p]]; p1r = DeleteCases[p1r, p]]]][[2,1]]];
Array[Delete[Map[plist, slist[#]], 0] &, 5]

A370219 Irregular triangle T(n,k) read by rows: row n lists run-length encoding d_k values (see comments) for the properly nested string of parentheses encoded by A063171(n).

Original entry on oeis.org

1, 1, 1, 0, 2, 1, 1, 1, 1, 0, 2, 0, 2, 1, 0, 1, 2, 0, 0, 3, 1, 1, 1, 1, 1, 1, 0, 2, 1, 0, 2, 1, 1, 0, 1, 2, 1, 0, 0, 3, 0, 2, 1, 1, 0, 2, 0, 2, 0, 1, 2, 1, 0, 1, 1, 2, 0, 1, 0, 3, 0, 0, 3, 1, 0, 0, 2, 2, 0, 0, 1, 3, 0, 0, 0, 4, 1, 1, 1, 1, 1, 1, 1, 1, 0, 2, 1, 1, 0, 2, 1

Offset: 1

Paolo Xausa, Feb 12 2024

As explained by Knuth (2011), a string of properly nested parentheses of length 2*m (for m >= 1) can be run-length encoded as ()d_1()d_2 ... ()d_m, where d_k are nonnegative integers such that d_1 + d_2 + ... + d_k <= k for 1 <= k < m and d_1 + d_2 + ... + d_m = m.

			The following table lists d_k values for properly nested strings having lengths up to 8, along with z_k, p_k and c_k values from related combinatorial objects (see related sequences for more information). Cf. Knuth (2011), p. 442, Table 1.
.
      | Properly |          |         | A370220 | A370221 | A370222
      | Nested   | A063171  | d d d d | z z z z | p p p p | c c c c
    n | String   |   (n)    | 1 2 3 4 | 1 2 3 4 | 1 2 3 4 | 1 2 3 4
  ----+----------+----------+---------+---------+---------+---------
    1 | ()       | 10       | 1       | 1       | 1       | 0
    2 | ()()     | 1010     | 1 1     | 1 3     | 1 2     | 0 0
    3 | (())     | 1100     | 0 2     | 1 2     | 2 1     | 0 1
    4 | ()()()   | 101010   | 1 1 1   | 1 3 5   | 1 2 3   | 0 0 0
    5 | ()(())   | 101100   | 1 0 2   | 1 3 4   | 1 3 2   | 0 0 1
    6 | (())()   | 110010   | 0 2 1   | 1 2 5   | 2 1 3   | 0 1 0
    7 | (()())   | 110100   | 0 1 2   | 1 2 4   | 2 3 1   | 0 1 1
    8 | ((()))   | 111000   | 0 0 3   | 1 2 3   | 3 2 1   | 0 1 2
    9 | ()()()() | 10101010 | 1 1 1 1 | 1 3 5 7 | 1 2 3 4 | 0 0 0 0
   10 | ()()(()) | 10101100 | 1 1 0 2 | 1 3 5 6 | 1 2 4 3 | 0 0 0 1
   11 | ()(())() | 10110010 | 1 0 2 1 | 1 3 4 7 | 1 3 2 4 | 0 0 1 0
   12 | ()(()()) | 10110100 | 1 0 1 2 | 1 3 4 6 | 1 3 4 2 | 0 0 1 1
   13 | ()((())) | 10111000 | 1 0 0 3 | 1 3 4 5 | 1 4 3 2 | 0 0 1 2
   14 | (())()() | 11001010 | 0 2 1 1 | 1 2 5 7 | 2 1 3 4 | 0 1 0 0
   15 | (())(()) | 11001100 | 0 2 0 2 | 1 2 5 6 | 2 1 4 3 | 0 1 0 1
   16 | (()())() | 11010010 | 0 1 2 1 | 1 2 4 7 | 2 3 1 4 | 0 1 1 0
   17 | (()()()) | 11010100 | 0 1 1 2 | 1 2 4 6 | 2 3 4 1 | 0 1 1 1
   18 | (()(())) | 11011000 | 0 1 0 3 | 1 2 4 5 | 2 4 3 1 | 0 1 1 2
   19 | ((()))() | 11100010 | 0 0 3 1 | 1 2 3 7 | 3 2 1 4 | 0 1 2 0
   20 | ((())()) | 11100100 | 0 0 2 2 | 1 2 3 6 | 3 2 4 1 | 0 1 2 1
   21 | ((()())) | 11101000 | 0 0 1 3 | 1 2 3 5 | 3 4 2 1 | 0 1 2 2
   22 | (((()))) | 11110000 | 0 0 0 4 | 1 2 3 4 | 4 3 2 1 | 0 1 2 3

Donald E. Knuth, The Art of Computer Programming, Vol. 4A: Combinatorial Algorithms, Part 1, Addison-Wesley, 2011, Section 7.2.1.6, pp. 440-444.

Paolo Xausa, Table of n, a(n) for n = 1..15521 (rows 1..2055 of the triangle, flattened).

Cf. A000108, A063171, A072643 (row lengths and row sums).

Cf. A370220, A370221, A370222.

zlist[m_] := With[{r = 2*Range[2, m]}, Reverse[Map[Join[{1}, #] &, Select[Subsets[Range[2, 2*m-1], {m-1}], Min[r-#] > 0 &]]]];
dlist[m_] := Map[Append[#, m - Total[#]] &, Map[Differences, zlist[m]] - 1];
Array[Delete[dlist[#], 0] &, 5]
(* 2nd program: uses Algorithm Z from Knuth's TAOCP section 7.2.1.6, exercise 2 *)
zlist[m_] := Block[{z = 2*Range[m] - 1, j},
    Reap[
    While[True,
        Sow[z];
        If[z[[m-1]] < z[[m]] - 1,
            z[[m]]--,
            j = m - 1; z[[m]] = 2*m - 1;
            While[j > 1 && z[[j-1]] == z[[j]] - 1, z[[j]] = 2*j - 1; j--];
            If[j == 1,Break[]];
            z[[j]]--]
    ]][[2]][[1]]];
dlist[m_] := Map[Append[#, m - Total[#]] &, Map[Differences, zlist[m]] - 1];
Join[{{1}}, Array[Delete[dlist[#], 0] &, 4, 2]] (* Paolo Xausa, Mar 25 2024 *)

T(n,k) = A370220(n,k+1) - A370220(n,k) - 1, for 1 <= k < A072643(n).

A371409 Irregular triangle T(n,k) read by rows: row n lists the positions of right parentheses in the properly nested string of parentheses encoded by A063171(n).

Original entry on oeis.org

2, 2, 4, 3, 4, 2, 4, 6, 2, 5, 6, 3, 4, 6, 3, 5, 6, 4, 5, 6, 2, 4, 6, 8, 2, 4, 7, 8, 2, 5, 6, 8, 2, 5, 7, 8, 2, 6, 7, 8, 3, 4, 6, 8, 3, 4, 7, 8, 3, 5, 6, 8, 3, 5, 7, 8, 3, 6, 7, 8, 4, 5, 6, 8, 4, 5, 7, 8, 4, 6, 7, 8, 5, 6, 7, 8, 2, 4, 6, 8, 10, 2, 4, 6, 9, 10, 2, 4, 7, 8, 10

Offset: 1

Paolo Xausa, Mar 22 2024

See A370220 for the positions of left parentheses.

			The following table lists the positions of right parentheses for properly nested strings having lengths up to 8, along with the positions of left parentheses.
.
      | Properly |          | Pos. of right | Pos. of left
      | Nested   | A063171  | parentheses   | parentheses
    n | String   |   (n)    | (this seq.)   | (A370220)
  ----+----------+----------+---------------+---------------
    1 | ()       | 10       | 2             | 1
    2 | ()()     | 1010     | 2 4           | 1 3
    3 | (())     | 1100     | 3 4           | 1 2
    4 | ()()()   | 101010   | 2 4 6         | 1 3 5
    5 | ()(())   | 101100   | 2 5 6         | 1 3 4
    6 | (())()   | 110010   | 3 4 6         | 1 2 5
    7 | (()())   | 110100   | 3 5 6         | 1 2 4
    8 | ((()))   | 111000   | 4 5 6         | 1 2 3
    9 | ()()()() | 10101010 | 2 4 6 8       | 1 3 5 7
   10 | ()()(()) | 10101100 | 2 4 7 8       | 1 3 5 6
   11 | ()(())() | 10110010 | 2 5 6 8       | 1 3 4 7
   12 | ()(()()) | 10110100 | 2 5 7 8       | 1 3 4 6
   13 | ()((())) | 10111000 | 2 6 7 8       | 1 3 4 5
   14 | (())()() | 11001010 | 3 4 6 8       | 1 2 5 7
   15 | (())(()) | 11001100 | 3 4 7 8       | 1 2 5 6
   16 | (()())() | 11010010 | 3 5 6 8       | 1 2 4 7
   17 | (()()()) | 11010100 | 3 5 7 8       | 1 2 4 6
   18 | (()(())) | 11011000 | 3 6 7 8       | 1 2 4 5
   19 | ((()))() | 11100010 | 4 5 6 8       | 1 2 3 7
   20 | ((())()) | 11100100 | 4 5 7 8       | 1 2 3 6
   21 | ((()())) | 11101000 | 4 6 7 8       | 1 2 3 5
   22 | (((()))) | 11110000 | 5 6 7 8       | 1 2 3 4

Donald E. Knuth, The Art of Computer Programming, Vol. 4A: Combinatorial Algorithms, Part 1, Addison-Wesley, 2011, Section 7.2.1.6, pp. 440-444.

Paolo Xausa, Table of n, a(n) for n = 1..15521 (rows 1..2055 of the triangle, flattened).

Cf. A063171, A370220, A072643 (row lengths), A371410 (row sums).

zlist[m_] := With[{r = 2*Range[2, m]}, Reverse[Map[Join[{1}, #] &, Select[Subsets[Range[2, 2*m-1], {m-1}], Min[r-#] > 0 &]]]];
Table[Delete[Map[Complement[Range[2*m], #] &, zlist[m]], 0], {m, 5}] (* Paolo Xausa, Mar 27 2024 *)
(* 2nd program: uses Algorithm Z from Knuth's TAOCP section 7.2.1.6, exercise 2 *)
zlist[m_] := Block[{z = 2*Range[m] - 1, j},
    Reap[
    While[True,
        Sow[z];
        If[z[[m-1]] < z[[m]] - 1,
            z[[m]]--,
            j = m - 1; z[[m]] = 2*m - 1;
            While[j > 1 && z[[j-1]] == z[[j]] - 1, z[[j]] = 2*j - 1; j--];
            If[j == 1,Break[]];
            z[[j]]--]
    ]][[2]][[1]]];
Join[{{2}}, Table[Delete[Map[Complement[Range[2*m], #] &, zlist[m]], 0], {m, 2, 5}]] (* Paolo Xausa, Mar 27 2024 *)

A371410 Row sums of A371409: sums of the positions of right parentheses in the properly nested string of parentheses encoded by A063171(n).

Original entry on oeis.org

2, 6, 7, 12, 13, 13, 14, 15, 20, 21, 21, 22, 23, 21, 22, 22, 23, 24, 23, 24, 25, 26, 30, 31, 31, 32, 33, 31, 32, 32, 33, 34, 33, 34, 35, 36, 31, 32, 32, 33, 34, 32, 33, 33, 34, 35, 34, 35, 36, 37, 33, 34, 34, 35, 36, 35, 36, 37, 38, 36, 37, 38, 39, 40, 42, 43, 43, 44, 45, 43

Offset: 1

Paolo Xausa, Mar 22 2024

nonn

See A370220 and A371409 for more information.

Donald E. Knuth, The Art of Computer Programming, Vol. 4A: Combinatorial Algorithms, Part 1, Addison-Wesley, 2011, Section 7.2.1.6, pp. 440-444.

Paolo Xausa, Table of n, a(n) for n = 1..10000

Cf. A063171, A370220, A370290, A371409.

zlist[m_] := With[{r = 2*Range[2, m]}, Reverse[Map[Join[{1}, #] &, Select[Subsets[Range[2, 2*m-1], {m-1}], Min[r-#] > 0 &]]]];
Table[Delete[Map[Total[Complement[Range[2*m], #]] &, zlist[m]], 0], {m, 5}] (* Paolo Xausa, Mar 25 2024 *)
(* 2nd program: uses Algorithm Z from Knuth's TAOCP section 7.2.1.6, exercise 2 *)
zlist[m_] := Block[{z = 2*Range[m] - 1, j},
    Reap[
    While[True,
        Sow[z];
        If[z[[m-1]] < z[[m]] - 1,
            z[[m]]--,
            j = m - 1; z[[m]] = 2*m - 1;
            While[j > 1 && z[[j-1]] == z[[j]] - 1, z[[j]] = 2*j - 1; j--];
            If[j == 1,Break[]];
            z[[j]]--]
    ]][[2]][[1]]];
Join[{2}, Table[Delete[Map[Total[Complement[Range[2*m], #]] &, zlist[m]], 0], {m, 2, 5}]] (* Paolo Xausa, Mar 25 2024 *)

cp's OEIS Frontend

A370290 Row sums of A370220.

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A370222 Irregular triangle T(n,k) read by rows: row n gives the inversion table (see comments) of the permutation encoded by row n of A370221.

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A370221 Irregular triangle T(n,k) read by rows: row n lists the values encoding a permutation (see comments) related to the properly nested string of parentheses encoded by A063171(n).

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A370219 Irregular triangle T(n,k) read by rows: row n lists run-length encoding d_k values (see comments) for the properly nested string of parentheses encoded by A063171(n).

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A371409 Irregular triangle T(n,k) read by rows: row n lists the positions of right parentheses in the properly nested string of parentheses encoded by A063171(n).

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A371410 Row sums of A371409: sums of the positions of right parentheses in the properly nested string of parentheses encoded by A063171(n).

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