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