A241652 Number of partitions p of n such that 2*(number of even numbers in p) <= (number of odd numbers in p).

1, 1, 1, 2, 2, 3, 5, 6, 10, 13, 21, 25, 40, 47, 69, 84, 117, 138, 187, 222, 292, 344, 439, 519, 654, 768, 951, 1118, 1378, 1612, 1968, 2308, 2807, 3282, 3977, 4657, 5630, 6585, 7936, 9278, 11170, 13046, 15648, 18274, 21868, 25481, 30402, 35385, 42069, 48875

Offset

0,4

Comments

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

Links

Andrew Howroyd, Table of n, a(n) for n = 0..1000

Formula

a(n) = A241651(n) + A241653(n) for n >= 0.

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

Example

a(6) counts these 5 partitions:  51, 33, 321, 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 *)

Prog

(PARI) seq(n)={my(p=prod(k=1, n, 1 + if(k%2, y, 1/y^2)*(x^k/(1 - x^k)), 1 + O(x*x^n))); Vec(sum(i=0, sqrtint(n), polcoef(p, i, y)))} \\ Andrew Howroyd, Dec 23 2025

Crossrefs

Cf. A241651, A241653, A241654, A241655.

Sequence in context: A160235 A227392 A050380 * A241636 A227360 A111077

Adjacent sequences: A241649 A241650 A241651 * A241653 A241654 A241655

Keyword

nonn,easy

Author

Clark Kimberling, Apr 27 2014

Status

approved