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

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

Author

Paolo Xausa, Mar 22 2024

Keywords

• nonn

Comments

See A370220 and A371409 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, pp. 440-444.

Links

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

Crossrefs

Cf. A063171, A370220, A370290, A371409.

Programs

Mathematica

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 *)