cp's OEIS Frontend

A330147 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).

%I A330147 #5 Feb 04 2020 14:57:19
%S A330147 0,1,2,3,4,4,8,8,15,20,30,40,63,78,110,143,190,238,313,389,501,621,
%T A330147 786,975,1231,1522,1901,2344,2930,3595,4451,5448,6700,8147,9974,12087,
%U A330147 14651,17672,21326,25558,30709,36657,43770,52069,61902,73357,86921,102697
%N A330147 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 A330147 For each partition of n, let
%C A330147 d = number of terms that are not repeated;
%C A330147 r = number of terms that are repeated.
%C A330147 a(n) is the number of partitions such that d != r.
%F A330147 a(n) + A241274(n) = A000041(n) for all n >= 0.
%e A330147 The partitions of 6 are 6, 51, 42, 411, 33, 321, 3111, 222, 2211, 21111, 111111.
%e A330147 These have d > r:  6, 51, 42, 321
%e A330147 These have d = r:  411, 3222, 21111
%e A330147 These have d < r:  33, 222, 2211, 111111
%e A330147 Thus, a(6) = 8
%t A330147 z = 30; d[p_] := Length[DeleteDuplicates[Select[p, Count[p, #] == 1 &]]];
%t A330147 r[p_] := Length[DeleteDuplicates[Select[p, Count[p, #] > 1 &]]]; Table[ Count[IntegerPartitions[n], p_ /; d[p] != r[p]], {n, 0, z}]
%Y A330147 Cf. A000041, A241274, A329976.
%K A330147 nonn,easy
%O A330147 0,3
%A A330147 _Clark Kimberling_, Feb 03 2020