A279385 Irregular triangle read by rows in which row n lists the numbers k such that the largest Dyck path of the symmetric representation of sigma(k) contains the point (n,n), or row n is 0 if no such k exists.

1, 2, 3, 4, 5, 0, 6, 7, 8, 9, 10, 11, 0, 12, 13, 14, 0, 15, 16, 17, 18, 19, 0, 20, 21, 22, 23, 0, 24, 25, 26, 27, 0, 28, 29, 0, 30, 31, 32, 33, 34, 0, 35, 36, 37, 38, 39, 0, 40, 41, 0, 42, 43, 44, 0, 45, 46, 47, 0, 48, 49, 50, 51, 52, 53, 0, 54, 55, 0, 56, 57, 58, 59, 0, 60, 61, 62, 0, 63, 64, 65, 0, 66, 67, 68, 69, 0

Offset

1,2

Comments

For more information about the mentioned Dyck paths see A237593.

Links

Table of n, a(n) for n=1..87.

Example

n         Triangle begins:
1         1;
2         2, 3;
3         4, 5;
4         0;
5         6, 7;
6         8,
7         9, 10, 11;
8         0;
9         12, 13, 14;
10        0;
11        15;
12        16, 17;
13        18, 19;
14        0;
15        20, 21, 22, 23;
16        0;
...

Mathematica

(* last computed value is dropped to avoid a potential under count of crossings *)
a240542[n_] := Sum[(-1)^(k+1)Ceiling[(n+1)/k-(k+1)/2], {k, 1, Floor[-1/2+1/2 Sqrt[8n+1]]}]
pathGroups[n_] := Module[{t}, t=Table[{}, a240542[n]]; Map[AppendTo[t[[a240542[#]]], #]&, Range[n]]; Map[If[t[[#]]=={}, t[[#]]={0}]&, Range[Length[t]]]; Most[t]]
a279385[n_] := Flatten[pathGroups[n]]
a279385[70] (* sequence *)
a279385T[n_] := TableForm[pathGroups[n], TableHeadings->{Range[a240542[n]-1], None}]
a279385T[24] (* display of irregular triangle - Hartmut F. W. Hoft, Feb 02 2022 *)

Crossrefs

Positive terms give A000027.

Cf. A259179(n) is the number of positive terms in row n.

Cf. A000203, A196020, A236104, A235791, A237048, A237591, A237593, A240542, A244050, A245092, A259179, A262626, A279286.

Sequence in context: A203572 A195829 A095874 * A267000 A365430 A224892

Adjacent sequences: A279382 A279383 A279384 * A279386 A279387 A279388

Keyword

nonn,tabf

Author

Omar E. Pol, Dec 12 2016

Extensions

More terms from Omar E. Pol, Jun 20 2018

Status

approved