A173306 Triangle read by rows, generated from an array of terms in powers of triangle A173305.

1, 1, 1, 1, 2, 1, 2, 2, 1, 3, 3, 1, 4, 5, 2, 5, 7, 3, 6, 10, 5, 1, 8, 14, 7, 1, 10, 19, 11, 2, 12, 26, 15, 3, 15, 35, 22, 5, 18, 46, 30, 7, 22, 60, 42, 11, 27, 78, 56, 15, 32, 10, 76, 22, 1, 38, 128, 100, 30, 1, 46, 162, 133, 42, 2, 54, 204, 173, 56, 3

Offset: 0

Author

Gary W. Adamson, Feb 15 2010

Keywords

• nonn
• tabf

Comments

Row sums = A000041, the partition numbers.

Examples

			Given triangle A173305, we create an array by extracting terms in powers of A173305:
1, 1, 1, 2, 2, 3, .4, .5, .6, .8, 10, 12, 15,...; = column terms of A173305
1, 1, 2, 3, 4, 6, .9, 12, 16, 22, 29, 38, 50,...; = terms of A173305^2
1, 1, 2, 3, 5, 7, 11, 15, 21, 29, 40, 53, 72,...; = terms of A173305^3
1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 56, 77,...; = terms of A173305^4
...
(rows quickly converge to A000041, the partition numbers).
Taking finite difference terms from the top, we obtain the array:
1, 1, 1, 2, 2, 3, .4, .5, .6,..8, 10, 12, 15,...;
......1, 1, 2, 3, .5, .7, 10, 14, 19, 26, 35,...;
............1, 1, .2, .3, .5, .7, 11, 15, 22,...;
...........................1, .1, .2, .3, .5,...;
...
Finally, columns of the above array become rows of A173306:
1;
1;
1, 1;
2, 1;
2, 2, 1;
3, 3, 1;
4, 5, 2;
5, 7, 3;
6, 10, 5, 1;
8, 14, 7, 1;
10, 19, 11, 2;
12, 26, 15, 3;
15, 35, 22, 5;
18, 46, 30, 7;
22, 60, 42, 11;
27, 78, 56, 15;
32, 100, 76, 22, 1;
38, 128, 100, 30, 1;
46, 162, 133, 42, 2;
54, 204, 173, 56, 3;
...
		

Crossrefs

Cf. A000009, A000041, A173305.

Formula

Given triangle A173305 in which every column >0 = A000009 shifted down twice.

We create an array in which n-th row = columns in (n-1)-th power of triangle

A173305. Finite differences of successive columns of the array become row terms

of A173306.