cp's OEIS Frontend

A259179 Number of Dyck paths described in A237593 that contain the point (n,n) in the diagram of the symmetric representation of sigma.

%I A259179 #101 Dec 31 2020 11:11:15
%S A259179 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,
%T A259179 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,
%U A259179 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
%N A259179 Number of Dyck paths described in A237593 that contain the point (n,n) in the diagram of the symmetric representation of sigma.
%C A259179 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.
%C A259179 a(n) is the number of occurrences of n in A240542. - _Omar E. Pol_, Dec 09 2016
%C A259179 Nonzero terms give A280919, the first differences of A071562. - _Omar E. Pol_, Apr 17 2018
%C A259179 Also first differences of A244367. Where records occur gives A279286. - _Omar E. Pol_, Apr 20 2020
%e A259179 Illustration of initial terms:
%e A259179 --------------------------------------------------------
%e A259179                            Diagram with 15 Dyck paths
%e A259179 n   A000203(n)  a(n)         to evaluate a(1)..a(10)
%e A259179 --------------------------------------------------------
%e A259179 .                         _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
%e A259179 1        1        1      |_| | | | | | | | | | | | | | |
%e A259179 2        3        2      |_ _|_| | | | | | | | | | | | |
%e A259179 3        4        2      |_ _|  _|_| | | | | | | | | | |
%e A259179 4        7        0      |_ _ _|    _|_| | | | | | | | |
%e A259179 5        6        2      |_ _ _|  _|  _ _|_| | | | | | |
%e A259179 6       12        1      |_ _ _ _|  _| |  _ _|_| | | | |
%e A259179 7        8        3      |_ _ _ _| |_ _|_|    _ _|_| | |
%e A259179 8       15        0      |_ _ _ _ _|  _|     |  _ _ _|_|
%e A259179 9       13        3      |_ _ _ _ _| |      _|_| |
%e A259179 10      18        0      |_ _ _ _ _ _|  _ _|    _|
%e A259179 .                        |_ _ _ _ _ _| |  _|  _|
%e A259179 .                        |_ _ _ _ _ _ _| |_ _|
%e A259179 .                        |_ _ _ _ _ _ _| |
%e A259179 .                        |_ _ _ _ _ _ _ _|
%e A259179 .                        |_ _ _ _ _ _ _ _|
%e A259179 .
%e A259179 For n = 3 there are two Dyck paths that contain the point (3,3) so a(3) = 2.
%e A259179 For n = 4 there are no Dyck paths that contain the point (4,4) so a(4) = 0.
%t A259179 a240542[n_] := Sum[(-1)^(k+1)Ceiling[(n+1)/k - (k+1)/2], {k, 1, Floor[(Sqrt[8n+1]-1)/2]}]
%t A259179 a259179[n_] := Module[{t=Table[0, n], k=1, d=1}, While[d<=n, t[[d]]+=1; d=a240542[++k]]; t] (* a(1..n) *)
%t A259179 a259179[102] (* _Hartmut F. W. Hoft_, Aug 06 2020 *)
%Y A259179 Cf. A000203, A024916, A071562, A196020, A235791, A236104, A237048, A237270, A237271, A237591, A237593, A240542, A244050, A244367, A244580, A245092, A249351, A256533, A259179, A262626, A279286, A280919.
%K A259179 nonn
%O A259179 1,2
%A A259179 _Omar E. Pol_, Aug 11 2015
%E A259179 More terms from _Omar E. Pol_, Dec 09 2016