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A082775 Convolution of natural numbers >= 2 and the partition numbers (A000041). 4

%I

%S 2,5,11,21,38,64,105,165,254,381,562,813,1162,1636,2279,3139,4285,

%T 5794,7776,10353,13694,17992,23502,30520,39433,50687,64855,82607,

%U 104785,132375,166608,208921,261090,325196,403779,499818,616928,759335,932135

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

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

%C 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.

%C Proof:

%C 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.

%C 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.

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

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

%F a(n) = a(n-1) + A000041(n) + A000070(n) for n>1. - _Alford Arnold_, Dec 10 2007

%F a(n) = n*A000070(n-2) - A182738(n-2) for n>2. - _Vaclav Kotesovec_, Jun 23 2015

%F a(n) ~ sqrt(3) * exp(Pi*sqrt(2*n/3)) / (2*Pi^2). - _Vaclav Kotesovec_, Jun 23 2015

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

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

%Y Cf. A023548, A126442.

%K easy,nonn

%O 2,1

%A _Alford Arnold_, May 22 2003

%E More terms from _Ray Chandler_, Oct 11 2003

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Last modified April 22 22:19 EDT 2021. Contains 343197 sequences. (Running on oeis4.)