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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.

%I A339575 #49 Jan 26 2022 08:59:35
%S A339575 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 A339575 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 A339575 7,6,6,6,12,12,12,12,12,12,12,10,10,9,7,7
%N 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.
%C A339575 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 A339575 Column k gives the partial sums of the k-th column of triangle A240061. - _Omar E. Pol_, Dec 11 2020
%H A339575 Rémy Sigrist, <a href="/A339575/b339575.txt">Table of n, a(n) for n = 1..10011</a> (rows for n = 1..141, flattened)
%H A339575 Rémy Sigrist, <a href="/A339575/a339575.gp.txt">PARI program for A339575</a>
%e A339575 Triangle begins:
%e A339575    1;
%e A339575    2,  2;
%e A339575    3,  3,  2;
%e A339575    4,  4,  4,  3;
%e A339575    5,  5,  5,  3,  3;
%e A339575    6,  6,  6,  6,  5,  4;
%e A339575    7,  7,  7,  7,  5,  4,  4;
%e A339575    8,  8,  8,  8,  8,  6,  5,  5;
%e A339575    9,  9,  9,  9,  9,  7,  7,  5,  5;
%e A339575   10, 10, 10, 10, 10, 10,  8,  7,  6,  6;
%e A339575   11, 11, 11, 11, 11, 11,  8,  7,  6,  6,  6;
%e A339575   12, 12, 12, 12, 12, 12, 12, 10, 10,  9,  7,  7;
%e A339575 ...
%e A339575 From _Omar E. Pol_, Jan 19 2022: (Start)
%e A339575 For n = 10 the Dyck path described in the 10th row of A237593 is as shown below in the fourth quadrant:
%e A339575 .
%e A339575    k                                     10th row
%e A339575       . . . . . . . . . . . . . X       of triangle
%e A339575    1  .                    |                10
%e A339575    2  .                    |                10
%e A339575    3  .                    |                10
%e A339575    4  .                    |                10
%e A339575    5  .                    |                10
%e A339575    6  .                 _ _|                10
%e A339575    7  .               _|                     8
%e A339575    8  .             _|                       7
%e A339575    9  .            |                         6
%e A339575   10  . _ _ _ _ _ _|                         6
%e A339575       .
%e A339575       .
%e A339575      -y
%e A339575 .
%e A339575 T(10,k) is the number of cells in the k-th row of the diagram.
%e A339575 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 A339575 (PARI) See Links section.
%Y A339575 Row sums are A024916.
%Y A339575 Cf. A000203, A236104, A237270, A237591, A237593, A240061.
%K A339575 nonn,tabl,look
%O A339575 1,2
%A A339575 _N. J. A. Sloane_, Dec 11 2020
%E A339575 Name edited by _Omar E. Pol_, Jan 22 2022