%I #43 Feb 05 2022 11:23:32

%S 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,

%T 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,

%U 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

%N 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.

%C For more information about the mentioned Dyck paths see A237593.

%e n Triangle begins:

%e 1 1;

%e 2 2, 3;

%e 3 4, 5;

%e 4 0;

%e 5 6, 7;

%e 6 8,

%e 7 9, 10, 11;

%e 8 0;

%e 9 12, 13, 14;

%e 10 0;

%e 11 15;

%e 12 16, 17;

%e 13 18, 19;

%e 14 0;

%e 15 20, 21, 22, 23;

%e 16 0;

%e ...

%t (* last computed value is dropped to avoid a potential under count of crossings *)

%t a240542[n_] := Sum[(-1)^(k+1)Ceiling[(n+1)/k-(k+1)/2], {k, 1, Floor[-1/2+1/2 Sqrt[8n+1]]}]

%t pathGroups[n_] := Module[{t}, t=Table[{}, a240542[n]]; Map[AppendTo[t[[a240542[#]]], #]&, Range[n]]; Map[If[t[[#]]=={}, t[[#]]={0}]&, Range[Length[t]]]; Most[t]]

%t a279385[n_] := Flatten[pathGroups[n]]

%t a279385[70] (* sequence *)

%t a279385T[n_] := TableForm[pathGroups[n], TableHeadings->{Range[a240542[n]-1], None}]

%t a279385T[24] (* display of irregular triangle - _Hartmut F. W. Hoft_, Feb 02 2022 *)

%Y Positive terms give A000027.

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

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

%K nonn,tabf

%O 1,2

%A _Omar E. Pol_, Dec 12 2016

%E More terms from _Omar E. Pol_, Jun 20 2018