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).

0, 1, 2, 3, 4, 4, 8, 8, 15, 20, 30, 40, 63, 78, 110, 143, 190, 238, 313, 389, 501, 621, 786, 975, 1231, 1522, 1901, 2344, 2930, 3595, 4451, 5448, 6700, 8147, 9974, 12087, 14651, 17672, 21326, 25558, 30709, 36657, 43770, 52069, 61902, 73357, 86921, 102697

Offset: 0

Author

Clark Kimberling, Feb 03 2020

Keywords

• nonn
• easy

Comments

For each partition of n, let

d = number of terms that are not repeated;

r = number of terms that are repeated.

a(n) is the number of partitions such that d != r.

Examples

			The partitions of 6 are 6, 51, 42, 411, 33, 321, 3111, 222, 2211, 21111, 111111.
These have d > r:  6, 51, 42, 321
These have d = r:  411, 3222, 21111
These have d < r:  33, 222, 2211, 111111
Thus, a(6) = 8
		

Crossrefs

Cf. A000041, A241274, A329976.

Programs

Mathematica

z = 30; d[p_] := Length[DeleteDuplicates[Select[p, Count[p, #] == 1 &]]];
r[p_] := Length[DeleteDuplicates[Select[p, Count[p, #] > 1 &]]]; Table[ Count[IntegerPartitions[n], p_ /; d[p] != r[p]], {n, 0, z}]

Formula

a(n) + A241274(n) = A000041(n) for all n >= 0.