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

%I A241637 #7 May 19 2014 10:43:08
%S A241637 1,1,1,3,3,7,8,14,16,26,32,45,57,78,103,132,174,220,295,361,477,584,
%T A241637 766,921,1194,1436,1841,2207,2782,3331,4169,4981,6156,7373,9019,10778,
%U A241637 13093,15636,18843,22507,26920,32096,38205,45470,53845,63970,75377,89356
%N A241637 Number of partitions p of n such that (number of even numbers in p) <= (number of odd numbers in p).
%C A241637 Each number in p is counted once, regardless of its multiplicity.
%H A241637 Alois P. Heinz, <a href="/A241637/b241637.txt">Table of n, a(n) for n = 0..1000</a>
%F A241637 a(n) = A241636(n) + A241638(n) for n >= 0.
%F A241637 a(n) + A241640(n) = A000041(n) for n >= 0.
%F A241637 a(n) = Sum_{k>=0} A242618(n,k). - _Alois P. Heinz_, May 19 2014
%e A241637 a(6) counts these 8 partitions:  51, 411, 33, 321, 3111, 2211, 21111, 111111.
%t A241637 z = 30; f[n_] := f[n] = IntegerPartitions[n]; s0[p_] := Count[Mod[DeleteDuplicates[p], 2],   0];
%t A241637 s1[p_] := Count[Mod[DeleteDuplicates[p], 2], 1];
%t A241637 Table[Count[f[n], p_ /; s0[p] < s1[p]], {n, 0, z}]  (* A241636 *)
%t A241637 Table[Count[f[n], p_ /; s0[p] <= s1[p]], {n, 0, z}] (* A241637 *)
%t A241637 Table[Count[f[n], p_ /; s0[p] == s1[p]], {n, 0, z}] (* A241638 *)
%t A241637 Table[Count[f[n], p_ /; s0[p] >= s1[p]], {n, 0, z}] (* A241639 *)
%t A241637 Table[Count[f[n], p_ /; s0[p] > s1[p]], {n, 0, z}]  (* A241640 *)
%Y A241637 Cf. A241636, A241638, A241639, A241640.
%K A241637 nonn,easy
%O A241637 0,4
%A A241637 _Clark Kimberling_, Apr 27 2014