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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.
8
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.
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.
Sequence in context: A203572 A195829 A095874 * A267000 A365430 A224892
KEYWORD
nonn,tabf
AUTHOR
Omar E. Pol, Dec 12 2016
EXTENSIONS
More terms from Omar E. Pol, Jun 20 2018
STATUS
approved