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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.

%I A279385 #43 Feb 05 2022 11:23:32
%S A279385 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 A279385 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 A279385 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 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.
%C A279385 For more information about the mentioned Dyck paths see A237593.
%e A279385 n         Triangle begins:
%e A279385 1         1;
%e A279385 2         2, 3;
%e A279385 3         4, 5;
%e A279385 4         0;
%e A279385 5         6, 7;
%e A279385 6         8,
%e A279385 7         9, 10, 11;
%e A279385 8         0;
%e A279385 9         12, 13, 14;
%e A279385 10        0;
%e A279385 11        15;
%e A279385 12        16, 17;
%e A279385 13        18, 19;
%e A279385 14        0;
%e A279385 15        20, 21, 22, 23;
%e A279385 16        0;
%e A279385 ...
%t A279385 (* last computed value is dropped to avoid a potential under count of crossings *)
%t A279385 a240542[n_] := Sum[(-1)^(k+1)Ceiling[(n+1)/k-(k+1)/2], {k, 1, Floor[-1/2+1/2 Sqrt[8n+1]]}]
%t A279385 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 a279385[n_] := Flatten[pathGroups[n]]
%t A279385 a279385[70] (* sequence *)
%t A279385 a279385T[n_] := TableForm[pathGroups[n], TableHeadings->{Range[a240542[n]-1], None}]
%t A279385 a279385T[24] (* display of irregular triangle - _Hartmut F. W. Hoft_, Feb 02 2022 *)
%Y A279385 Positive terms give A000027.
%Y A279385 Cf. A259179(n) is the number of positive terms in row n.
%Y A279385 Cf. A000203, A196020, A236104, A235791, A237048, A237591, A237593, A240542, A244050, A245092, A259179, A262626, A279286.
%K A279385 nonn,tabf
%O A279385 1,2
%A A279385 _Omar E. Pol_, Dec 12 2016
%E A279385 More terms from _Omar E. Pol_, Jun 20 2018