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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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified April 17 09:29 EDT 2021. Contains 343064 sequences. (Running on oeis4.)