A210765 Triangle read by rows in which row n lists the number of partitions of n together with n-1 ones.

1, 2, 1, 3, 1, 1, 5, 1, 1, 1, 7, 1, 1, 1, 1, 11, 1, 1, 1, 1, 1, 15, 1, 1, 1, 1, 1, 1, 22, 1, 1, 1, 1, 1, 1, 1, 30, 1, 1, 1, 1, 1, 1, 1, 1, 42, 1, 1, 1, 1, 1, 1, 1, 1, 1, 56, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 77, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 101, 1, 1, 1

Offset: 1

Author

Omar E. Pol, Mar 26 2012

Keywords

• nonn
• tabl
• easy

Comments

The sum of row n is S_n = n - 1 + A000041(n) = A133041(n) - 1.

Also consider a vertical rectangle on the infinite square grid with shorter side = n and longer side = p(n) = A000041(n). Each row of rectangle represents a partition of n. Each part of each partition of n is a horizontal rectangle with shorter side = 1 and longer side = k, where k is the size of the part. It appears that T(n,k) is also the number of k-th parts of all partitions of n in the k-th column of rectangle.

Examples

			Triangle begins:
1;
2,  1;
3,  1, 1;
5,  1, 1, 1;
7,  1, 1, 1, 1;
11, 1, 1, 1, 1, 1;
15, 1, 1, 1, 1, 1, 1;
22, 1, 1, 1, 1, 1, 1, 1;
30, 1, 1, 1, 1, 1, 1, 1, 1;
42, 1, 1, 1, 1, 1, 1, 1, 1, 1;
		

Crossrefs

Main diagonal of A209655 and of A209918.

Cf. A000041, A008284, A058399, A133041, A135010, A141285, A209656, A207380.