A241637 Number of partitions p of n such that (number of even numbers in p) <= (number of odd numbers in p).

1, 1, 1, 3, 3, 7, 8, 14, 16, 26, 32, 45, 57, 78, 103, 132, 174, 220, 295, 361, 477, 584, 766, 921, 1194, 1436, 1841, 2207, 2782, 3331, 4169, 4981, 6156, 7373, 9019, 10778, 13093, 15636, 18843, 22507, 26920, 32096, 38205, 45470, 53845, 63970, 75377, 89356

Offset

0,4

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) = A241636(n) + A241638(n) for n >= 0.

a(n) + A241640(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 8 partitions:  51, 411, 33, 321, 3111, 2211, 21111, 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_ /; 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, A241638, A241639, A241640.

Sequence in context: A200792 A218567 A161416 * A241641 A339398 A241414

Adjacent sequences: A241634 A241635 A241636 * A241638 A241639 A241640

Keyword

nonn,easy

Author

Clark Kimberling, Apr 27 2014

Status

approved