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A239242 Number of partitions of n into distinct parts for which (number of odd parts) > (number of even parts).

%I A239242 #16 Aug 30 2016 03:50:57
%S A239242 0,1,0,1,1,1,2,1,4,2,6,3,9,5,12,9,17,14,22,22,29,33,38,48,50,68,65,95,
%T A239242 86,128,113,172,149,226,197,295,260,379,342,485,449,613,587,773,762,
%U A239242 967,987,1206,1269,1497,1623,1855,2063,2289,2610,2823,3280,3471
%N A239242 Number of partitions of n into distinct parts for which (number of odd parts) > (number of even parts).
%C A239242 a(n) = Sum_{k>=1} A240021(n,k). - _Alois P. Heinz_, Apr 02 2014
%H A239242 Alois P. Heinz, <a href="/A239242/b239242.txt">Table of n, a(n) for n = 0..1000</a>
%F A239242 a(n) + A239240(n) = A000009(n) for n >=1.
%e A239242 a(8) = 4 counts these partitions:  71, 53, 521, 431.
%p A239242 b:= proc(n, i, t) option remember; `if`(n>i*(i+1)/2, 0,
%p A239242      `if`(n=0, `if`(t>0, 1, 0 ), b(n, i-1, t)+`if`(i>n, 0,
%p A239242       b(n-i, i-1, t+`if`(irem(i, 2)=1, 1, -1)))))
%p A239242     end:
%p A239242 a:= n-> b(n$2, 0):
%p A239242 seq(a(n), n=0..60);  # _Alois P. Heinz_, Mar 15 2014
%t A239242 z = 55; p[n_] := p[n] = IntegerPartitions[n]; d[u_] := d[u] = DeleteDuplicates[u]; g[u_] := g[u] = Length[u];
%t A239242 Table[g[Select[Select[p[n], d[#] == # &], Count[#, _?OddQ] < Count[#, _?EvenQ] &]], {n, 0, z}] (* A239239 *)
%t A239242 Table[g[Select[Select[p[n], d[#] == # &], Count[#, _?OddQ] <= Count[#, _?EvenQ] &]], {n, 0, z}] (* A239240 *)
%t A239242 Table[g[Select[Select[p[n], d[#] == # &], Count[#, _?OddQ] == Count[#, _?EvenQ] &]], {n, 0, z}] (* A239241 *)
%t A239242 Table[g[Select[Select[p[n], d[#] == # &], Count[#, _?OddQ] > Count[#, _?EvenQ] &]], {n, 0, z}] (* A239242 *)
%t A239242 Table[g[Select[Select[p[n], d[#] == # &], Count[#, _?OddQ] >= Count[#, _?EvenQ] &]], {n, 0, z}] (* A239243 *)
%t A239242 (* _Peter J. C. Moses_, Mar 10 2014 *)
%t A239242 b[n_, i_, t_] := b[n, i, t] = If[n>i*(i+1)/2, 0, If[n==0, If[t>0, 1, 0], b[n, i-1, t]+If[i>n, 0, b[n-i, i-1, t+If[Mod[i, 2]==1, 1, -1]]]]]; a[n_] := b[n, n, 0]; Table[a[n], {n, 0, 60}] (* _Jean-François Alcover_, Aug 30 2016, after _Alois P. Heinz_ *)
%Y A239242 Cf. A239239, A239240, A239241, A239243, A000009.
%K A239242 nonn,easy
%O A239242 0,7
%A A239242 _Clark Kimberling_, Mar 13 2014