A207383 Triangle read by rows: T(n,k) is the sum of parts of size k in the last section of the set of partitions of n.

1, 1, 2, 2, 0, 3, 3, 4, 0, 4, 5, 2, 3, 0, 5, 7, 8, 6, 4, 0, 6, 11, 6, 6, 4, 5, 0, 7, 15, 16, 9, 12, 5, 6, 0, 8, 22, 14, 18, 8, 10, 6, 7, 0, 9, 30, 30, 18, 20, 15, 12, 7, 8, 0, 10, 42, 30, 30, 20, 20, 12, 14, 8, 9, 0, 11, 56, 54, 42, 40, 25, 30, 14, 16, 9, 10, 0, 12

Offset: 1

Author

Omar E. Pol, Feb 24 2012

Keywords

• nonn
• tabl

Comments

For further properties of this triangle see also A182703.

Examples

			Triangle begins:
   1;
   1,  2;
   2,  0,  3;
   3,  4,  0,  4;
   5,  2,  3,  0,  5;
   7,  8,  6,  4,  0,  6;
  11,  6,  6,  4,  5,  0,  7;
  15, 16,  9, 12,  5,  6,  0,  8;
  22, 14, 18,  8, 10,  6,  7,  0,  9;
  30, 30, 18, 20, 15, 12,  7,  8,  0, 10;
  42, 30, 30, 20, 20, 12, 14,  8,  9,  0, 11;
  56, 54, 42, 40, 25, 30, 14, 16,  9, 10,  0, 12;
...
From _Omar E. Pol_, Nov 28 2020: (Start)
Illustration of three arrangements of the last section of the set of partitions of 7, or more generally the 7th section of the set of partitions of any integer >= 7:
.                                        _ _ _ _ _ _ _
.     (7)                    (7)        |_ _ _ _      |
.     (4+3)                (4+3)        |_ _ _ _|_    |
.     (5+2)                (5+2)        |_ _ _    |   |
.     (3+2+2)            (3+2+2)        |_ _ _|_ _|_  |
.       (1)                  (1)                    | |
.         (1)                (1)                    | |
.         (1)                (1)                    | |
.           (1)              (1)                    | |
.         (1)                (1)                    | |
.           (1)              (1)                    | |
.           (1)              (1)                    | |
.             (1)            (1)                    | |
.             (1)            (1)                    | |
.               (1)          (1)                    | |
.                 (1)        (1)                    |_|
.    ----------------
.     19,8,5,3,2,1,1 --> Row 7 of triangle A207031
.      |/|/|/|/|/|/|
.     11,3,2,1,1,0,1 --> Row 7 of triangle A182703
.      * * * * * * *
.      1,2,3,4,5,6,7 --> Row 7 of triangle A002260
.      = = = = = = =
.     11,6,6,4,5,0,7 --> Row 7 of this triangle
.
Note that the "head" of the last section is formed by the partitions of 7 that do not contain 1 as a part. The "tail" is formed by A000041(7-1) parts of size 1. The number of rows (or zones) is A000041(7) = 15. The last section of the set of partitions of 7 contains eleven 1's, three 2's, two 3's, one 4, one 5, there are no 6's and it contains one 7. So the 7th row of triangle is [11, 6, 6, 4, 5, 0, 7]. (End)
		

Links

• Alois P. Heinz, Rows n = 1..141, flattened

Crossrefs

Column 1 is A000041.

Leading diagonal gives A000027.

Second diagonal gives A000007.

Row sums give A138879.

Cf. A002260, A066186, A135010, A138121, A138785, A138880, A182703, A194812, A207031.

Formula

T(n,k) = k*A182703(n,k).