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

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

Offset: 1

Author

Omar E. Pol, Aug 11 2015

Keywords

• nonn

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

Examples

			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.
		

Crossrefs

Cf. A000203, A024916, A071562, A196020, A235791, A236104, A237048, A237270, A237271, A237591, A237593, A240542, A244050, A244367, A244580, A245092, A249351, A256533, A259179, A262626, A279286, A280919.

Programs

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 *)

Extensions

More terms from Omar E. Pol, Dec 09 2016