A112531 Triangle read by rows which lists compositions having at least one part equal to 1.

1, 1, 2, 1, 2, 3, 3, 1, 2, 3, 6, 4, 3, 6, 4, 1, 2, 3, 6, 4, 6, 12, 12, 5, 3, 6, 12, 10, 4, 10, 5, 1, 2, 3, 6, 4, 6, 12, 12, 5, 6, 12, 24, 20, 12, 30, 20, 6, 3, 6, 12, 10, 12, 30, 30, 15, 4, 10, 20, 20, 5, 15, 6, 1

Offset: 1

Author

Alford Arnold, Sep 10 2005

Keywords

• nonn
• tabf

Comments

Consider partitions listed in the order given by A241596 and A242628. Omit any partition not containing 1 as a part. Write down the number of compositions (= ordered partitions) corresponding to this partition.

Row sums give A112532; which are the first differences of A047970.

Row lengths give A011782.

Examples

			The partitions (see A241596) begin 1 2 11 3 22 21 111 4 33 32 222 31 221 211 1111 ...
After omitting partitions with no part equal to 1, we have
1 11 21 111 31 221 211 1111 ...
which give rise to 1 1 2 1 2 3 3 1 ... compositions.
The resulting triangle of compositions begins:
1;
1;
2, 1;
2, 3, 3, 1;
2, 3, 6, 4, 3, 6, 4, 1;
2, 3, 6, 4, 6, 12, 12, 5, 3, 6, 12, 10, 4, 10, 5, 1;
2, 3, 6, 4, 6, 12, 12, 5, 6, 12, 24, 20, 12, 30, 20, 6, 3, 6, 12, 10, 12, 30, 30, 15, 4, 10, 20, 20, 5, 15, 6, 1;
...
		

Crossrefs

Cf. A000079, A011782, A047970, A089246, A112532, A241596, A242628.

Extensions

Edited by N. J. A. Sloane, May 19 2014 based on postings to the Sequence Fans Mailing List by Peter Luschny, Jonas Wallgren, Arie Groeneveld, and Franklin T. Adams-Watters.