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 A259179 Number of Dyck paths described in A237593 that contain the point (n,n) in the diagram of the symmetric representation of sigma. 17
 1, 2, 2, 0, 2, 1, 3, 0, 3, 0, 1, 2, 2, 0, 4, 0, 1, 3, 0, 2, 0, 2, 3, 0, 1, 4, 0, 2, 0, 3, 0, 3, 0, 1, 1, 4, 0, 2, 0, 4, 0, 3, 0, 1, 2, 0, 4, 0, 2, 0, 0, 5, 0, 3, 0, 1, 3, 0, 4, 0, 2, 0, 1, 0, 5, 0, 2, 1, 0, 1, 4, 0, 4, 0, 2, 0, 2, 0, 5, 0, 3, 0, 0, 0, 1, 5, 0, 2, 2, 0, 2, 0, 3, 0, 5, 0, 3, 0, 1, 0, 0, 6 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS Since the diagram of the symmetric representation of sigma is also the top view of the stepped pyramid described in A245092, and the diagram is also the top view of the staircase described in A244580, so we have that a(n) is also the height difference (or length of the vertical line segment) at the point (n,n) in the main diagonal of the mentioned structures. a(n) is the number of occurrences of n in A240542. - Omar E. Pol, Dec 09 2016 Nonzero terms give A280919, the first differences of A071562. - Omar E. Pol, Apr 17 2018 Also first differences of A244367. Where records occur gives A279286. - Omar E. Pol, Apr 20 2020 LINKS EXAMPLE Illustration of initial terms: --------------------------------------------------------                            Diagram with 15 Dyck paths n   A000203(n)  a(n)         to evaluate a(1)..a(10) -------------------------------------------------------- .                         _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 1        1        1      |_| | | | | | | | | | | | | | | 2        3        2      |_ _|_| | | | | | | | | | | | | 3        4        2      |_ _|  _|_| | | | | | | | | | | 4        7        0      |_ _ _|    _|_| | | | | | | | | 5        6        2      |_ _ _|  _|  _ _|_| | | | | | | 6       12        1      |_ _ _ _|  _| |  _ _|_| | | | | 7        8        3      |_ _ _ _| |_ _|_|    _ _|_| | | 8       15        0      |_ _ _ _ _|  _|     |  _ _ _|_| 9       13        3      |_ _ _ _ _| |      _|_| | 10      18        0      |_ _ _ _ _ _|  _ _|    _| .                        |_ _ _ _ _ _| |  _|  _| .                        |_ _ _ _ _ _ _| |_ _| .                        |_ _ _ _ _ _ _| | .                        |_ _ _ _ _ _ _ _| .                        |_ _ _ _ _ _ _ _| . For n = 3 there are two Dyck paths that contain the point (3,3) so a(3) = 2. For n = 4 there are no Dyck paths that contain the point (4,4) so a(4) = 0. MATHEMATICA a240542[n_] := Sum[(-1)^(k+1)Ceiling[(n+1)/k - (k+1)/2], {k, 1, Floor[(Sqrt[8n+1]-1)/2]}] a259179[n_] := Module[{t=Table[0, n], k=1, d=1}, While[d<=n, t[[d]]+=1; d=a240542[++k]]; t] (* a(1..n) *) a259179[102] (* Hartmut F. W. Hoft, Aug 06 2020 *) CROSSREFS Cf. A000203, A024916, A071562, A196020, A235791, A236104, A237048, A237270, A237271, A237591, A237593, A240542, A244050, A244367, A244580, A245092, A249351, A256533, A259179, A262626, A279286, A280919. Sequence in context: A236374 A045719 A114906 * A214667 A214665 A229723 Adjacent sequences:  A259176 A259177 A259178 * A259180 A259181 A259182 KEYWORD nonn AUTHOR Omar E. Pol, Aug 11 2015 EXTENSIONS More terms from Omar E. Pol, Dec 09 2016 STATUS approved

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Last modified May 11 09:11 EDT 2021. Contains 343788 sequences. (Running on oeis4.)