A236002 Number of overcompositions of n.

1, 2, 4, 12, 26, 60, 144, 324, 728, 1602, 3576, 7808, 17068, 36908, 79520, 170704, 364794, 777036, 1649456, 3491188, 7367544, 15513336, 32584648, 68307264, 142904080, 298448914, 622235060, 1295320004, 2692583916, 5589586996, 11588905844, 23999052692

Offset

0,2

Comments

Analog to overpartitions, here an overcomposition is defined to be a composition in which the first occurrence of each distinct number may be overlined (see example).

Also 1 together with the row sums of A235999.

For the number of partitions of n see A000041.

For the number of compositions of n see A011782.

For the number of overpartitions of n see A015128.

Note that there are several orderings of overcompositions, the same as the orderings of compositions, but apparently for every ordering of overcompositions there are also several suborderings according to the arrangements of the overlined parts. The same for overpartitions. See one of them in Example section.

Links

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Formula

a(n) = Sum_{k=1..A003056(n)} 2^k*A235998(n,k), n >= 1.

Example

For n = 4 the 26 overcompositions of 4 are: [4], [4'], [1,3], [1',3], [1,3'], [1',3'], [2,2], [2',2], [1,1,2], [1',1,2], [1,1,2'], [1',1,2'], [3,1], [3',1], [3,1'], [3',1'], [1,2,1], [1',2,1], [1,2',1], [1',2',1], [2,1,1], [2',1,1], [2,1',1], [2',1',1], [1,1,1,1], [1',1,1,1].

Crossrefs

Cf. A000041, A011782, A015128, A235998, A235999, A238439.

Sequence in context: A266079 A284125 A045678 * A291406 A219752 A242536

Adjacent sequences: A235999 A236000 A236001 * A236003 A236004 A236005

Keyword

nonn

Author

Omar E. Pol, Jan 19 2014

Extensions

a(7) corrected and more terms added, Joerg Arndt, Jan 20 2014

a(19)-a(31) from Alois P. Heinz, Jan 20 2014

Status

approved