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