A188139 Triangle by rows, A027293 * A129372 as infinite lower triangular matrices.

1, 1, 1, 3, 1, 1, 4, 2, 1, 1, 8, 3, 2, 1, 1, 11, 6, 3, 2, 1, 1, 19, 8, 5, 3, 2, 1, 1, 26, 13, 7, 5, 3, 2, 1, 1, 41, 18, 12, 7, 5, 3, 2, 1, 1, 56, 28, 16, 11, 7, 5, 3, 2, 1, 1, 83, 38, 24, 15, 11, 7, 5, 3, 2, 1, 1, 112, 55, 33, 23, 15, 11, 7, 5, 3, 2, 1, 1

Offset: 1

Author

Gary W. Adamson, Mar 21 2011

Keywords

• nonn
• tabl

Comments

Row sums = A066897: (1, 2, 5, 8, 15, 24, 39,...), total number of odd parts in all partitions of n.

Apparently T(n,k) is the number of (2*k)'s in all the partitions of (n+k), k>=1, e.g. T(7,3) = number of 6's in partitions of 10 = A024790(10). [David Scambler, May 24 2012]

Examples

			First few rows of the triangle =
.
1,
1, 1
3, 1, 1
4, 2, 1, 1
8, 3, 2, 1, 1
11, 6, 3, 2, 1, 1
19, 8, 5, 3, 2, 1, 1
26, 13, 7, 5, 3, 2, 1, 1
41, 18, 12, 7, 5, 3, 2, 1, 1
56, 28, 16, 11, 7, 5, 3, 2, 1, 1
83, 38, 24, 15, 11, 7, 5, 3, 2, 1, 1
112, 55, 33, 23, 15, 11, 7, 5, 3, 2, 1, 1
160, 74, 47, 31, 22, 15, 11, 7, 5, 3, 2, 1, 1,
...
		

Crossrefs

Cf. A027293, A066897, A129372.

Programs

Mathematica

Table[Count[Flatten[IntegerPartitions[n+k]], 2*k], {n,1,15}, {k,1,n}] (* David Scambler, May 24 2012 *)

Extensions

a(22) ff. corrected and more terms from Georg Fischer, Jun 10 2023