cp's OEIS Frontend

A317304 Numbers k with the property that both Dyck paths of the symmetric representation of sigma(k) have a central valley.

%I A317304 #47 Sep 14 2018 10:57:24
%S A317304 4,5,11,12,13,14,22,23,24,25,26,27,37,38,39,40,41,42,43,44,56,57,58,
%T A317304 59,60,61,62,63,64,65,79,80,81,82,83,84,85,86,87,88,89,90,106,107,108,
%U A317304 109,110,111,112,113,114,115,116,117,118,119,137,138,139,140,141,142,143,144,145,146,147,148,149
%N A317304 Numbers k with the property that both Dyck paths of the symmetric representation of sigma(k) have a central valley.
%C A317304 Also triangle read by rows which gives the even-indexed rows of triangle A014132.
%C A317304 There are no triangular number (A000217) in this sequence.
%C A317304 For more information about the symmetric representation of sigma see A237593 and its related sequences.
%C A317304 Equivalently, numbers k with the property that both Dyck paths of the symmetric representation of sigma(k) have an even number of peaks. - _Omar E. Pol_, Sep 13 2018
%e A317304 Written as an irregular triangle in which the row lengths are the positive even numbers, the sequence begins:
%e A317304     4,   5;
%e A317304    11,  12,  13,  14;
%e A317304    22,  23,  24,  25,  26,  27;
%e A317304    37,  38,  39,  40,  41,  42,  43,  44;
%e A317304    56,  57,  58,  59,  60,  61,  62,  63,  64,  65;
%e A317304    79,  80,  81,  82,  83,  84,  85,  86,  87,  88,  89,  90;
%e A317304   106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119;
%e A317304 ...
%e A317304 Illustration of initial terms:
%e A317304 -------------------------------------------------
%e A317304    k  sigma(k)  Diagram of the symmetry of sigma
%e A317304 -------------------------------------------------
%e A317304                        _ _           _ _ _ _
%e A317304                       | | |         | | | | |
%e A317304                      _| | |         | | | | |
%e A317304                  _ _|  _|_|         | | | | |
%e A317304    4      7     |_ _ _|             | | | | |
%e A317304    5      6     |_ _ _|             | | | | |
%e A317304                                  _ _|_| | | |
%e A317304                                _|    _ _|_| |
%e A317304                              _|     |  _ _ _|
%e A317304                             |      _|_|
%e A317304                  _ _ _ _ _ _|  _ _|
%e A317304   11     12     |_ _ _ _ _ _| |  _|
%e A317304   12     28     |_ _ _ _ _ _ _| |
%e A317304   13     14     |_ _ _ _ _ _ _| |
%e A317304   14     24     |_ _ _ _ _ _ _ _|
%e A317304 .
%e A317304 For the first six terms of the sequence we can see in the above diagram that both Dyck path (the smallest and the largest) of the symmetric representation of sigma(k) have a central valley.
%e A317304 Compare with A317303.
%Y A317304 Row sums give A084367. n >= 1.
%Y A317304 Column 1 gives A084849, n >= 1.
%Y A317304 Column 2 gives A096376, n >= 1.
%Y A317304 Right border gives the nonzero terms of A014106.
%Y A317304 The union of A000217, A317303 and this sequence gives A001477.
%Y A317304 Some other sequences related to the central peak or the central valley of the symmetric representation of sigma are A000217, A000384, A007606, A007607, A014105, A014132, A162917, A161983, A317303. See also A317306.
%Y A317304 Cf. A000203, A005843, A196020, A236104, A235791, A237048, A237591, A237593, A237270, A237271, A239660, A239931, A239932, A239933, A239934, A244050, A245092, A249351, A262626.
%K A317304 nonn,tabf
%O A317304 1,1
%A A317304 _Omar E. Pol_, Aug 27 2018