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

1,1

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

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 *)

Crossrefs

Cf. A063171, A370220, A370290, A371409.

Sequence in context: A226814 A233419 A189327 * A217575 A172154 A293531

Adjacent sequences: A371407 A371408 A371409 * A371411 A371412 A371413

Keyword

nonn

Author

Paolo Xausa, Mar 22 2024

Status

approved