A241638 Number of partitions p of n such that (number of even numbers in p) = (number of odd numbers in p).

1, 0, 0, 1, 1, 4, 3, 8, 6, 13, 11, 20, 17, 31, 34, 47, 56, 78, 103, 125, 167, 203, 281, 315, 433, 487, 673, 745, 989, 1101, 1472, 1623, 2116, 2386, 3052, 3430, 4347, 4948, 6168, 7104, 8673, 10068, 12210, 14234, 17047, 20007, 23671, 27869, 32739, 38609, 45010

Offset

0,6

Comments

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

Links

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Formula

a(n) = A241637(n) - A241636(n) = A241639(n) - A241640(n) for n >= 0.

a(n) + A241636(n) + A241640(n) = A000041(n) for n >= 0.

a(n) = A242618(n,0). - Alois P. Heinz, May 19 2014

Example

a(6) counts these 3 partitions:  411, 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_ /; s0[p] < s1[p]], {n, 0, z}]  (* A241636 *)
Table[Count[f[n], p_ /; s0[p] <= s1[p]], {n, 0, z}] (* A241637 *)
Table[Count[f[n], p_ /; s0[p] == s1[p]], {n, 0, z}] (* A241638 *)
Table[Count[f[n], p_ /; s0[p] >= s1[p]], {n, 0, z}] (* A241639 *)
Table[Count[f[n], p_ /; s0[p] > s1[p]], {n, 0, z}]  (* A241640 *)

Crossrefs

Cf. A241636, A241637, A241639, A241640.

Sequence in context: A200089 A330686 A117956 * A360689 A110662 A265289

Adjacent sequences: A241635 A241636 A241637 * A241639 A241640 A241641

Keyword

nonn,easy

Author

Clark Kimberling, Apr 27 2014

Status

approved