|
%I #5 Feb 04 2020 14:57:19
%S 0,1,2,3,4,4,8,8,15,20,30,40,63,78,110,143,190,238,313,389,501,621,
%T 786,975,1231,1522,1901,2344,2930,3595,4451,5448,6700,8147,9974,12087,
%U 14651,17672,21326,25558,30709,36657,43770,52069,61902,73357,86921,102697
%N Number of partitions p of n such that (number of numbers in p that have multiplicity 1) != (number of numbers in p having multiplicity > 1).
%C For each partition of n, let
%C d = number of terms that are not repeated;
%C r = number of terms that are repeated.
%C a(n) is the number of partitions such that d != r.
%F a(n) + A241274(n) = A000041(n) for all n >= 0.
%e The partitions of 6 are 6, 51, 42, 411, 33, 321, 3111, 222, 2211, 21111, 111111.
%e These have d > r: 6, 51, 42, 321
%e These have d = r: 411, 3222, 21111
%e These have d < r: 33, 222, 2211, 111111
%e Thus, a(6) = 8
%t z = 30; d[p_] := Length[DeleteDuplicates[Select[p, Count[p, #] == 1 &]]];
%t r[p_] := Length[DeleteDuplicates[Select[p, Count[p, #] > 1 &]]]; Table[ Count[IntegerPartitions[n], p_ /; d[p] != r[p]], {n, 0, z}]
%Y Cf. A000041, A241274, A329976.
%K nonn,easy
%O 0,3
%A _Clark Kimberling_, Feb 03 2020
|
