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

Author

Omar E. Pol, Dec 12 2016

Keywords

• nonn
• tabf

Comments

For more information about the mentioned Dyck paths see A237593.

Examples

			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;
...
		

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.

Programs

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 *)

Extensions

More terms from Omar E. Pol, Jun 20 2018