A208482 Triangle read by rows: T(n,k) = sum of positive k-th ranks of all partitions of n.

0, 1, 1, 2, 1, 1, 4, 1, 2, 1, 7, 1, 3, 2, 1, 12, 2, 5, 4, 2, 1, 18, 3, 6, 6, 4, 2, 1, 29, 6, 9, 10, 7, 4, 2, 1, 42, 9, 11, 13, 11, 7, 4, 2, 1, 63, 16, 15, 19, 17, 12, 7, 4, 2, 1, 89, 24, 18, 25, 24, 18, 12, 7, 4, 2, 1, 128, 39, 24, 36, 34, 28, 19, 12, 7, 4, 2, 1

Offset: 1

Author

Omar E. Pol, Mar 07 2012

Keywords

• nonn
• tabl

Comments

For the definition of the k-th rank see A208478.

It appears that the sum of the k-th ranks of all partitions of n is equal to zero.

It appears that reversed rows converge to A000070, the same as A208478. - Omar E. Pol, Mar 10 2012

Examples

			For n = 4 the partitions of 4 and the four types of ranks of the partitions of 4 are
----------------------------------------------------------
Partitions    First      Second       Third      Fourth
of 4          rank        rank        rank        rank
----------------------------------------------------------
4           4-1 =  3    0-1 = -1    0-1 = -1    0-1 = -1
3+1         3-2 =  1    1-1 =  0    0-1 = -1    0-0 =  0
2+2         2-2 =  0    2-2 =  0    0-0 =  0    0-0 =  0
2+1+1       2-3 = -1    1-1 =  0    1-0 =  1    0-0 =  0
1+1+1+1     1-4 = -3    1-0 =  1    1-0 =  1    1-0 =  1
----------------------------------------------------------
The sums of positive k-th ranks of the partitions of 4 are 4, 1, 2, 1 so row 4 lists 4, 1, 2, 1.
Triangle begins:
0;
1,    1;
2,    1,  1;
4,    1,  2,  1;
7,    1,  3,  2,  1;
12,   2,  5,  4,  2,  1;
18,   3,  6,  6,  4,  2,  1;
29,   6,  9, 10,  7,  4,  2,  1;
42,   9, 11, 13, 11,  7,  4,  2,  1;
63,  16, 15, 19, 17, 12,  7,  4,  2,  1;
89,  24, 18, 25, 24, 18, 12,  7,  4,  2,  1;
128, 39, 24, 36, 34, 28, 19, 12,  7,  4,  2,  1;
		

Links

• Alois P. Heinz, Rows n = 1..44, flattened

Crossrefs

Column 1 is A209616. Row sums give A208483.

Cf. A006128, A063995, A064173, A105805, A181187, A194547, A194549, A195822, A208478.

Extensions

Terms a(1)-a(22) confirmed and additional terms added by John W. Layman, Mar 10 2012