cp's OEIS Frontend

A330145 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 A330145 #6 Feb 04 2020 14:57:14
%S A330145 1,1,1,2,3,6,7,13,16,24,30,43,52,73,91,122,154,204,258,335,423,545,
%T A330145 684,865,1081,1348,1675,2073,2546,3123,3821,4648,5656,6851,8282,9966,
%U A330145 12031,14416,17315,20695,24754,29477,35170,41738,49638,58735,69613,82119
%N A330145 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 A330145 For each partition of n, let
%C A330145 d = number of terms that are not repeated;
%C A330145 r = number of terms that are repeated.
%C A330145 a(n) is the number of partitions such that d >= r.
%F A330145 a(n) + A330001(n) = A000041(n) for all n >= 0.
%e A330145 The partitions of 6 are 6, 51, 42, 411, 33, 321, 3111, 222, 2211, 21111, 111111.
%e A330145 These have d > r:  6, 51, 42, 321
%e A330145 These have d = r:  411, 3222, 21111
%e A330145 These have d < r:  33, 222, 2211, 111111
%e A330145 Thus, a(6) = 7
%t A330145 z = 30; d[p_] := Length[DeleteDuplicates[Select[p, Count[p, #] == 1 &]]];
%t A330145 r[p_] := Length[DeleteDuplicates[Select[p, Count[p, #] > 1 &]]]; Table[Count[IntegerPartitions[n], p_ /; d[p] >=  r[p]], {n, 0, z}]
%Y A330145 Cf. A000041, A241274, A329976.
%K A330145 nonn,easy
%O A330145 0,4
%A A330145 _Clark Kimberling_, Feb 03 2020