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A082775
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Convolution of natural numbers >= 2 and the partition numbers (A000041).
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4
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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; internal format)
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OFFSET
| 2,1
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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.
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FORMULA
| a(n)= a(n-1)+A000041(n) + A000070(n) for n>1 - Alford Arnold (Alford1940(AT)aol.com), Dec 10 2007
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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.
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MATHEMATICA
| f[n_] := Sum[(n - k) PartitionsP[k], {k, 0, n - 2}]; Array[f, 39, 2]
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CROSSREFS
| Cf. A023548, A126442.
Sequence in context: A003522 A112805 A119970 * A023548 A144700 A000785
Adjacent sequences: A082772 A082773 A082774 * A082776 A082777 A082778
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KEYWORD
| easy,nonn,changed
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AUTHOR
| Alford Arnold (Alford1940(AT)aol.com), May 22 2003
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EXTENSIONS
| More terms from Ray Chandler (rayjchandler(AT)sbcglobal.net), Oct 11 2003
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