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A241651
Number of partitions p of n such that 2*(number of even numbers in p) < (number of odd numbers in p).
5
0, 1, 1, 2, 2, 3, 4, 5, 6, 8, 10, 13, 16, 22, 27, 38, 47, 66, 81, 112, 136, 187, 227, 301, 363, 473, 568, 727, 862, 1088, 1289, 1596, 1876, 2304, 2697, 3265, 3810, 4583, 5327, 6358, 7354, 8736, 10101, 11924, 13750, 16195, 18653, 21883, 25177, 29484, 33906
OFFSET
0,4
COMMENTS
Each number in p is counted once, regardless of its multiplicity.
FORMULA
a(n) = A241652(n) - A241653(n) for n >= 0.
a(n) + A241653(n) + A241655(n) = A000041(n) for n >= 0.
EXAMPLE
a(6) counts these 4 partitions: 51, 33, 3111, 111111.
MATHEMATICA
z = 30; f[n_] := f[n] = IntegerPartitions[n]; s0[p_] := Count[Mod[DeleteDuplicates[p], 2], 0];
s1[p_] := Count[Mod[DeleteDuplicates[p], 2], 1];
Table[Count[f[n], p_ /; 2 s0[p] < s1[p]], {n, 0, z}] (* A241651 *)
Table[Count[f[n], p_ /; 2 s0[p] <= s1[p]], {n, 0, z}] (* A241652 *)
Table[Count[f[n], p_ /; 2 s0[p] == s1[p]], {n, 0, z}] (* A241653 *)
Table[Count[f[n], p_ /; 2 s0[p] >= s1[p]], {n, 0, z}] (* A241654 *)
Table[Count[f[n], p_ /; 2 s0[p] > s1[p]], {n, 0, z}] (* A241655 *)
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Apr 27 2014
STATUS
approved