%I #10 Feb 06 2014 09:54:39
%S 1,2,4,8,16,27,48,75,118,178,265,377,544,760,1048,1437,1949,2611,3480,
%T 4594,6024,7867,10184,13122,16823,21484,27258,34495,43425,54499,68105,
%U 84870,105322,130412,160832,197932,242776,297145,362535,441464,536064,649703
%N Number of distinct cyclic permutations of the partitions of n; see comments.
%C Suppose that p = [x(1),...,x(k)], is a partition of n, where x(1) <= x(2) <= ... <= x(k). If x(1) = x(k), there is only one cyclic permutation of p; otherwise, there are k of them.
%H Alois P. Heinz, <a href="/A236292/b236292.txt">Table of n, a(n) for n = 1..400</a>
%F a(n) = (d(n), f(2), f(3),..., f(n-1))*(1,2,3,...,n-1), where d(n) = (number of divisors of n) = (number of constant partitions of n), and f(k) = number of nonconstant partitions of n, for k = 2,3,...,n-1.
%e a(6) = (4,2,2,2,1)*(1,2,3,4,5) = 27, where * = convolution. The 5 components of (4,2,2,2,1) count these partitions: (6, 33, 222, 1111); (51, 42); (411, 321); (3111, 2211); (211111).
%t Map[Total[Map[Length, Map[(# /. Table[x_, {Length[#]}] -> {x}) &, IntegerPartitions[#]]]] &, Range[40]] (* A236292 *)
%t (* _Peter J. C. Moses_, Jan 21 2014 *)
%Y Cf. A236293.
%K nonn,easy
%O 1,2
%A _Clark Kimberling_, Jan 22 2014