cp's OEIS Frontend

A330001 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 A330001 #4 Feb 04 2020 14:57:07
%S A330001 0,0,1,1,2,1,4,2,6,6,12,13,25,28,44,54,77,93,127,155,204,247,318,390,
%T A330001 494,610,761,937,1172,1442,1783,2194,2693,3292,4028,4917,5946,7221,
%U A330001 8700,10490,12584,15106,18004,21523,25537,30399,35945,42635,50219,59382
%N A330001 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 A330001 For each partition of n, let
%C A330001 d = number of terms that are not repeated;
%C A330001 r = number of terms that are repeated.
%C A330001 a(n) is the number of partitions such that d < r.
%F A330001 a(n) + A241274(n) + A329976(n) = A000041(n) for n >= 0.
%e A330001 The partitions of 6 are 6, 51, 42, 411, 33, 321, 3111, 222, 2211, 21111, 111111.
%e A330001 These have d > r:  6, 51, 42, 321
%e A330001 These have d = r:  411, 3222, 21111
%e A330001 These have d < r:  33, 222, 2211, 111111
%e A330001 Thus, a(6) = 4.
%t A330001 z = 30; d[p_] := Length[DeleteDuplicates[Select[p, Count[p, #] == 1 &]]];
%t A330001 r[p_] := Length[DeleteDuplicates[Select[p, Count[p, #] > 1 &]]]; Table[Count[IntegerPartitions[n], p_ /; d[p] <  r[p]], {n, 0, z}]
%Y A330001 Cf. A000041, A241274, A329976.
%K A330001 nonn,easy
%O A330001 0,5
%A A330001 _Clark Kimberling_, Feb 03 2020