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

1,3

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.

Links

Table of n, a(n) for n=1..64.

Example

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.

Sequence in context: A118977 A071766 A007305 * A373553 A100002 A376238

Adjacent sequences: A112528 A112529 A112530 * A112532 A112533 A112534

Keyword

nonn,tabf

Author

Alford Arnold, Sep 10 2005

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.

Status

approved