A241639 Number of partitions p of n such that (number of even numbers in p) >= (number of odd numbers in p).

1, 0, 1, 1, 3, 4, 6, 9, 12, 17, 21, 31, 37, 54, 66, 91, 113, 155, 193, 254, 317, 411, 517, 649, 814, 1009, 1268, 1548, 1925, 2335, 2907, 3484, 4309, 5156, 6343, 7535, 9231, 10949, 13340, 15782, 19091, 22555, 27179, 32025, 38377, 45171, 53852, 63267, 75076

Offset

0,5

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) = A241638(n) + A241640(n) for n >= 0.

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

a(n) = Sum_{k<=0} A242618(n,k). - Alois P. Heinz, May 19 2014

Example

a(6) counts these 6 partitions:  6, 42, 411, 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_ /; 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, A241638, A241640.

Sequence in context: A182531 A097922 A103109 * A241655 A288346 A078743

Adjacent sequences: A241636 A241637 A241638 * A241640 A241641 A241642

Keyword

nonn,easy

Author

Clark Kimberling, Apr 27 2014

Status

approved