A241654 Number of partitions p of n such that 2*(number of even numbers in p) = (number of odd numbers in p).

1, 0, 1, 1, 3, 4, 7, 10, 16, 22, 32, 43, 61, 79, 108, 138, 184, 231, 304, 378, 491, 605, 775, 954, 1212, 1485, 1868, 2283, 2856, 3477, 4315, 5246, 6473, 7839, 9613, 11618, 14167, 17054, 20688, 24827, 29984, 35847, 43073, 51337, 61425, 72939, 86905, 102871

Offset

0,5

Comments

Each number in p is counted once, regardless of its multiplicity.

Links

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

Formula

a(n) = A241653(n) + A241655(n) for n >= 0.

a(n) + A241651(n) + A241655(n) = A000041(n) for n >= 0.

Example

a(6) counts these 7 partitions:  6, 42, 411, 321, 222, 2211, 21111.

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

Cf. A241651, A241652, A241653, A241655.

Sequence in context: A275633 A004397 A324368 * A047967 A282893 A256912

Adjacent sequences: A241651 A241652 A241653 * A241655 A241656 A241657

Keyword

nonn,easy

Author

Clark Kimberling, Apr 27 2014

Status

approved