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

Author

Omar E. Pol, Jan 19 2014

Keywords

• nonn

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.

Examples

			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].
		

Links

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

Crossrefs

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

Formula

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

Extensions

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

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