

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



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
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OFFSET

0,3


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.


LINKS

Table of n, a(n) for n=0..47.


FORMULA

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


EXAMPLE

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


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}]


CROSSREFS

Cf. A000041, A241274, A329976.
Sequence in context: A319079 A325329 A224038 * A241037 A097093 A056877
Adjacent sequences: A330144 A330145 A330146 * A330148 A330149 A330150


KEYWORD

nonn,easy


AUTHOR

Clark Kimberling, Feb 03 2020


STATUS

approved



