A339575 - OEIS

%I #49 Jan 26 2022 08:59:35

%S 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,

%T 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,

%U 7,6,6,6,12,12,12,12,12,12,12,10,10,9,7,7

%N 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.

%C 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).

%C Column k gives the partial sums of the k-th column of triangle A240061. - _Omar E. Pol_, Dec 11 2020

%H Rémy Sigrist, <a href="/A339575/b339575.txt">Table of n, a(n) for n = 1..10011</a> (rows for n = 1..141, flattened)

%H Rémy Sigrist, <a href="/A339575/a339575.gp.txt">PARI program for A339575</a>

%e Triangle begins:

%e    1;

%e    2,  2;

%e    3,  3,  2;

%e    4,  4,  4,  3;

%e    5,  5,  5,  3,  3;

%e    6,  6,  6,  6,  5,  4;

%e    7,  7,  7,  7,  5,  4,  4;

%e    8,  8,  8,  8,  8,  6,  5,  5;

%e    9,  9,  9,  9,  9,  7,  7,  5,  5;

%e   10, 10, 10, 10, 10, 10,  8,  7,  6,  6;

%e   11, 11, 11, 11, 11, 11,  8,  7,  6,  6,  6;

%e   12, 12, 12, 12, 12, 12, 12, 10, 10,  9,  7,  7;

%e ...

%e From _Omar E. Pol_, Jan 19 2022: (Start)

%e For n = 10 the Dyck path described in the 10th row of A237593 is as shown below in the fourth quadrant:

%e .

%e    k                                     10th row

%e       . . . . . . . . . . . . . X       of triangle

%e    1  .                    |                10

%e    2  .                    |                10

%e    3  .                    |                10

%e    4  .                    |                10

%e    5  .                    |                10

%e    6  .                 _ _|                10

%e    7  .               _|                     8

%e    8  .             _|                       7

%e    9  .            |                         6

%e   10  . _ _ _ _ _ _|                         6

%e       .

%e       .

%e      -y

%e .

%e T(10,k) is the number of cells in the k-th row of the diagram.

%e 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)

%o (PARI) See Links section.

%Y Row sums are A024916.

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

%K nonn,tabl,look

%O 1,2

%A _N. J. A. Sloane_, Dec 11 2020

%E Name edited by _Omar E. Pol_, Jan 22 2022