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A082775 Convolution of natural numbers >= 2 and the partition numbers (A000041). 4
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 (list; graph; refs; listen; history; text; internal format)
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.

Sequence in context: A003522 A112805 A119970 * 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

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Last modified March 28 02:40 EDT 2017. Contains 284182 sequences.