A240061 -id:A240061 - cp's OEIS frontend

A339575 Triangle read by rows: T(n,k) is the number of cells in the k-th row of the diagram constructed in the fourth quadrant with the Dyck path described in the n-th row of A237593, 1 <= k <= n.

1, 2, 2, 3, 3, 2, 4, 4, 4, 3, 5, 5, 5, 3, 3, 6, 6, 6, 6, 5, 4, 7, 7, 7, 7, 5, 4, 4, 8, 8, 8, 8, 8, 6, 5, 5, 9, 9, 9, 9, 9, 7, 7, 5, 5, 10, 10, 10, 10, 10, 10, 8, 7, 6, 6, 11, 11, 11, 11, 11, 11, 8, 7, 6, 6, 6, 12, 12, 12, 12, 12, 12, 12, 10, 10, 9, 7, 7

Offset: 1

N. J. A. Sloane, Dec 11 2020

These are the lengths of the rows if one regards the n-th region in the diagram as the Young diagram corresponding to a partition of A024916(n).

Column k gives the partial sums of the k-th column of triangle A240061. - Omar E. Pol, Dec 11 2020

			Triangle begins:
   1;
   2,  2;
   3,  3,  2;
   4,  4,  4,  3;
   5,  5,  5,  3,  3;
   6,  6,  6,  6,  5,  4;
   7,  7,  7,  7,  5,  4,  4;
   8,  8,  8,  8,  8,  6,  5,  5;
   9,  9,  9,  9,  9,  7,  7,  5,  5;
  10, 10, 10, 10, 10, 10,  8,  7,  6,  6;
  11, 11, 11, 11, 11, 11,  8,  7,  6,  6,  6;
  12, 12, 12, 12, 12, 12, 12, 10, 10,  9,  7,  7;
...
From _Omar E. Pol_, Jan 19 2022: (Start)
For n = 10 the Dyck path described in the 10th row of A237593 is as shown below in the fourth quadrant:
.
   k                                     10th row
      . . . . . . . . . . . . . X       of triangle
   1  .                    |                10
   2  .                    |                10
   3  .                    |                10
   4  .                    |                10
   5  .                    |                10
   6  .                 _ _|                10
   7  .               _|                     8
   8  .             _|                       7
   9  .            |                         6
  10  . _ _ _ _ _ _|                         6
      .
      .
     -y
.
T(10,k) is the number of cells in the k-th row of the diagram.
The total number of cells in all rows of the diagram is equal to A024916(10) = 87, the same as the sum of the 10th row of triangle. (End)

Rémy Sigrist, Table of n, a(n) for n = 1..10011 (rows for n = 1..141, flattened)
Rémy Sigrist, PARI program for A339575

Row sums are A024916.

Cf. A000203, A236104, A237270, A237591, A237593, A240061.

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Name edited by Omar E. Pol, Jan 22 2022

A240060 Triangle read by rows, n>=1, 1<=k<=n. T(n,n-k+1) = number of cells in the k-th row = number of cells in the k-th column of the diagram of the symmetric representation of sigma(n) in the first quadrant.

Original entry on oeis.org

1, 2, 1, 2, 1, 1, 3, 2, 1, 1, 3, 0, 1, 1, 1, 4, 2, 3, 1, 1, 1, 4, 0, 0, 1, 1, 1, 1, 5, 1, 2, 3, 1, 1, 1, 1, 5, 0, 2, 1, 1, 1, 1, 1, 1, 6, 1, 2, 1, 3, 1, 1, 1, 1, 1, 6, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 7, 1, 3, 4, 3, 4, 1, 1, 1, 1, 1, 1, 7, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Omar E. Pol, Apr 26 2014

Since the diagram is symmetric the number of cells in the k-th row equals the number of cell in the k-th column, see example.
Row sums give A000203.
Column 1 gives A008619, n >= 1.
If n is an odd prime then row n lists (n+1)/2, ((n+1)/2 - 2) zeros, and (n+1)/2 ones.
Mirror of A240061.

			Triangle begins:
1;
2, 1;
2, 1, 1;
3, 2, 1, 1;
3, 0, 1, 1, 1;
4, 2, 3, 1, 1, 1;
4, 0, 0, 1, 1, 1, 1;
5, 1, 2, 3, 1, 1, 1, 1;
5, 0, 2, 1, 1, 1, 1, 1, 1;
6, 1, 2, 1, 3, 1, 1, 1, 1, 1;
6, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1;
7, 1, 3, 4, 3, 4, 1, 1, 1, 1, 1, 1;
7, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1;
...
For n = 9 the symmetric representation of sigma(9) = 13 in the first quadrant looks like this:
y
.                               Number of cells
._ _ _ _ _
|_ _ _ _ _|                            5
.         |_ _                         0
.         |_  |                        2
.           |_|_ _                     1
.               | |                    1
.               | |                    1
.               | |                    1
.               | |                    1
. . . . . . . . |_| . . x              1
.
So the 9th row of triangle is [5, 0, 2, 1, 1, 1, 1, 1, 1].
For n = 9 and k = 7 there are two cells in the 7th row of the diagram, also there are two cells in the 7th column of the diagram, so T(9,9-7+1) = T(9,3) = 2.

Index entries for sequences related to sigma(n)

Cf. A000203, A008619, A024916, A196020, A236104, A235791, A237270, A237271, A237591, A237593, A239660, A239931-A239934, A240061.

cp's OEIS Frontend

A339575 Triangle read by rows: T(n,k) is the number of cells in the k-th row of the diagram constructed in the fourth quadrant with the Dyck path described in the n-th row of A237593, 1 <= k <= n.

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