A082775 Convolution of natural numbers >= 2 and the partition numbers (A000041).

2, 5, 11, 21, 38, 64, 105, 165, 254, 381, 562, 813, 1162, 1636, 2279, 3139, 4285, 5794, 7776, 10353, 13694, 17992, 23502, 30520, 39433, 50687, 64855, 82607, 104785, 132375, 166608, 208921, 261090, 325196, 403779, 499818, 616928, 759335, 932135

Offset

2,1

Comments

Contribution from George Beck, Jan 08 2011: (Start)

The number of multiset partitions of the n-multiset M={0,0,...,0,1,2} (with n-2 zeros) is sum_{k=0..(n-2)}( (n-k) * p(k) ) where p(k) is the number of partitions of k.

Proof:

For each k = 0, 1, ..., n-2, partition k zeros and add the remaining n-k-2 zeros to the block {1, 2}, to give p(k) partitions.

For each k, partition k zeros and add the remaining n-k-2 zeros to the two blocks {1} and {2} in all possible 1 + n-k-2 ways, which gives (1 + n-k-2) * p(k) partitions.

Together, the number of partitions of M is sum_{k=0..n-2}( (n-k) * p(k) ). (End)

A082775 is the special case of A126442 with n-k = 2.

Links

Table of n, a(n) for n=2..40.

Formula

a(n) = a(n-1) + A000041(n) + A000070(n) for n>1. - Alford Arnold, Dec 10 2007

a(n) = n*A000070(n-2) - A182738(n-2) for n>2. - Vaclav Kotesovec, Jun 23 2015

a(n) ~ sqrt(3) * exp(Pi*sqrt(2*n/3)) / (2*Pi^2). - Vaclav Kotesovec, Jun 23 2015

Example

a(7) = 64 because (7,5,3,2,1,1) dot (2,3,4,5,6,7) = 14+15+12+10+6+7= 64.

Mathematica

f[n_] := Sum[(n - k) PartitionsP[k], {k, 0, n - 2}]; Array[f, 39, 2]

Crossrefs

Cf. A023548, A126442, A346822.

Column k=2 of A346426.

Sequence in context: A112805 A119970 A326509 * A023548 A144700 A000785

Adjacent sequences: A082772 A082773 A082774 * A082776 A082777 A082778

Keyword

easy,nonn

Author

Alford Arnold, May 22 2003

Extensions

More terms from Ray Chandler, Oct 11 2003

Status

approved