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A241651 Number of partitions p of n such that 2*(number of even numbers in p) < (number of odd numbers in p). 5

%I

%S 0,1,1,2,2,3,4,5,6,8,10,13,16,22,27,38,47,66,81,112,136,187,227,301,

%T 363,473,568,727,862,1088,1289,1596,1876,2304,2697,3265,3810,4583,

%U 5327,6358,7354,8736,10101,11924,13750,16195,18653,21883,25177,29484,33906

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

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

%F a(n) = A241652(n) - A241653(n) for n >= 0.

%F a(n) + A241653(n) + A241655(n) = A000041(n) for n >= 0.

%e a(6) counts these 4 partitions: 51, 33, 3111, 111111.

%t z = 30; f[n_] := f[n] = IntegerPartitions[n]; s0[p_] := Count[Mod[DeleteDuplicates[p], 2], 0];

%t s1[p_] := Count[Mod[DeleteDuplicates[p], 2], 1];

%t Table[Count[f[n], p_ /; 2 s0[p] < s1[p]], {n, 0, z}] (* A241651 *)

%t Table[Count[f[n], p_ /; 2 s0[p] <= s1[p]], {n, 0, z}] (* A241652 *)

%t Table[Count[f[n], p_ /; 2 s0[p] == s1[p]], {n, 0, z}] (* A241653 *)

%t Table[Count[f[n], p_ /; 2 s0[p] >= s1[p]], {n, 0, z}] (* A241654 *)

%t Table[Count[f[n], p_ /; 2 s0[p] > s1[p]], {n, 0, z}] (* A241655 *)

%Y Cf. A241652, A241653, A241654, A241655.

%K nonn,easy

%O 0,4

%A _Clark Kimberling_, Apr 27 2014

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Last modified October 26 13:09 EDT 2021. Contains 348267 sequences. (Running on oeis4.)