cp's OEIS Frontend

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

%I A082775 #30 Aug 06 2021 15:23:06
%S A082775 2,5,11,21,38,64,105,165,254,381,562,813,1162,1636,2279,3139,4285,
%T A082775 5794,7776,10353,13694,17992,23502,30520,39433,50687,64855,82607,
%U A082775 104785,132375,166608,208921,261090,325196,403779,499818,616928,759335,932135
%N A082775 Convolution of natural numbers >= 2 and the partition numbers (A000041).
%C A082775 Contribution from _George Beck_, Jan 08 2011: (Start)
%C A082775 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 A082775 Proof:
%C A082775 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 A082775 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 A082775 Together, the number of partitions of M is sum_{k=0..n-2}( (n-k) * p(k) ). (End)
%C A082775 A082775 is the special case of A126442 with n-k = 2.
%F A082775 a(n) = a(n-1) + A000041(n) + A000070(n) for n>1. - _Alford Arnold_, Dec 10 2007
%F A082775 a(n) = n*A000070(n-2) - A182738(n-2) for n>2. - _Vaclav Kotesovec_, Jun 23 2015
%F A082775 a(n) ~ sqrt(3) * exp(Pi*sqrt(2*n/3)) / (2*Pi^2). - _Vaclav Kotesovec_, Jun 23 2015
%e A082775 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 A082775 f[n_] := Sum[(n - k) PartitionsP[k], {k, 0, n - 2}]; Array[f, 39, 2]
%Y A082775 Cf. A023548, A126442, A346822.
%Y A082775 Column k=2 of A346426.
%K A082775 easy,nonn
%O A082775 2,1
%A A082775 _Alford Arnold_, May 22 2003
%E A082775 More terms from _Ray Chandler_, Oct 11 2003