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A241636 Number of partitions p of n such that (number of even numbers in p) < (number of odd numbers in p).

%I A241636 #8 May 19 2014 10:43:34
%S A241636 0,1,1,2,2,3,5,6,10,13,21,25,40,47,69,85,118,142,192,236,310,381,485,
%T A241636 606,761,949,1168,1462,1793,2230,2697,3358,4040,4987,5967,7348,8746,
%U A241636 10688,12675,15403,18247,22028,25995,31236,36798,43963,51706,61487,72197
%N A241636 Number of partitions p of n such that (number of even numbers in p) < (number of odd numbers in p).
%C A241636 Each number in p is counted once, regardless of its multiplicity.
%H A241636 Alois P. Heinz, <a href="/A241636/b241636.txt">Table of n, a(n) for n = 0..1000</a>
%F A241636 a(n) + A241638(n) = A241637(n) for n >= 0.
%F A241636 a(n) + A241638(n) + A241640(n) = A000041(n) for n >= 0.
%F A241636 a(n) = Sum_{k>0} A242618(n,k). - _Alois P. Heinz_, May 19 2014
%e A241636 a(6) counts these 5 partitions:  51, 33, 321, 3111, 111111.
%t A241636 z = 30; f[n_] := f[n] = IntegerPartitions[n]; s0[p_] := Count[Mod[DeleteDuplicates[p], 2],   0];
%t A241636 s1[p_] := Count[Mod[DeleteDuplicates[p], 2], 1];
%t A241636 Table[Count[f[n], p_ /; s0[p] < s1[p]], {n, 0, z}]  (* A241636 *)
%t A241636 Table[Count[f[n], p_ /; s0[p] <= s1[p]], {n, 0, z}] (* A241637 *)
%t A241636 Table[Count[f[n], p_ /; s0[p] == s1[p]], {n, 0, z}] (* A241638 *)
%t A241636 Table[Count[f[n], p_ /; s0[p] >= s1[p]], {n, 0, z}] (* A241639 *)
%t A241636 Table[Count[f[n], p_ /; s0[p] > s1[p]], {n, 0, z}]  (* A241640 *)
%Y A241636 Cf. A241637, A241638, A241639, A241640.
%K A241636 nonn,easy
%O A241636 0,4
%A A241636 _Clark Kimberling_, Apr 27 2014