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

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

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 *)

Prog

(PARI) a(n)={my(s=0); forpart(p=n, my(S=Set(p), k=#select(x->x%2==0, S)); if(3*k>=#S, s++)); s} \\ Andrew Howroyd, Mar 27 2025
(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(subst(p, y, 1)-sum(i=1, sqrtint(n), polcoef(p, i, y)))} \\ Andrew Howroyd, Dec 23 2025

Crossrefs

Cf. A241651, A241652, A241653, A241655.

Sequence in context: A275633 A004397 A324368 * A047967 A282893 A385009

Adjacent sequences: A241651 A241652 A241653 * A241655 A241656 A241657

Keyword

nonn,easy

Author

Clark Kimberling, Apr 27 2014

Extensions

Name corrected by Andrew Howroyd, Mar 27 2025

Status

approved