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A209423 Difference between the number of odd parts and the number of even parts in all the partitions of n.

%I A209423 #44 Aug 05 2020 21:56:50
%S A209423 1,1,4,4,10,13,24,30,52,68,105,137,202,264,376,485,669,864,1162,1486,
%T A209423 1968,2501,3256,4110,5285,6630,8434,10511,13241,16417,20505,25273,
%U A209423 31344,38438,47346,57782,70746,85947,104663,126594,153386,184793,222865,267452
%N A209423 Difference between the number of odd parts and the number of even parts in all the partitions of n.
%C A209423 a(n) = number of parts of odd multiplicity (each counted only once) in all partitions of n. Example: a(5) = 10 because we have [5'],[4',1'],[3',2'], [3',1,1],[2,2,1'],[2',1',1,1], and [1',1,1,1,1] (the 10 counted parts are marked). - _Emeric Deutsch_, Feb 08 2016
%H A209423 Vaclav Kotesovec, <a href="/A209423/b209423.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..1000 from Alois P. Heinz)
%F A209423 a(n) = A066897(n) - A066898(n) = A206563(n,1) - A206563(n,2). - _Omar E. Pol_, Mar 08 2012
%F A209423 G.f.: Sum_{j>0} x^j/(1+x^j)/Product_{k>0}(1 - x^k). - _Emeric Deutsch_, Feb 08 2016
%F A209423 a(n) = Sum_{i=1..n} (-1)^(i + 1)*A181187(n, i). - _John M. Campbell_, Mar 18 2018
%F A209423 a(n) ~ log(2) * exp(Pi*sqrt(2*n/3)) / (2^(3/2) * Pi * sqrt(n)). - _Vaclav Kotesovec_, May 25 2018
%F A209423 For n > 0, a(n) = A305121(n) + A305123(n). - _Vaclav Kotesovec_, May 26 2018
%F A209423 a(n) = Sum_{k=-floor(n/2)+(n mod 2)..n} k * A240009(n,k). - _Alois P. Heinz_, Oct 23 2018
%F A209423 a(n) = Sum_{k>0} k * A264398(n,k). - _Alois P. Heinz_, Aug 05 2020
%e A209423 The partitions of 5 are [5], [4,1], [3,2], [3,1,1], [2,2,1], [2,1,1,1], and [1,1,1,1,1], a total of 15 odd parts and 5 even parts, so that a(5)=10.
%p A209423 b:= proc(n, i) option remember; local m, f, g;
%p A209423       m:= irem(i, 2);
%p A209423       if n=0 then [1, 0, 0]
%p A209423     elif i<1 then [0, 0, 0]
%p A209423     else f:= b(n, i-1); g:= `if`(i>n, [0$3], b(n-i, i));
%p A209423          [f[1]+g[1], f[2]+g[2]+m*g[1], f[3]+g[3]+(1-m)*g[1]]
%p A209423       fi
%p A209423     end:
%p A209423 a:= n-> b(n, n)[2] -b(n, n)[3]:
%p A209423 seq(a(n), n=1..50);  # _Alois P. Heinz_, Jul 09 2012
%p A209423 g := add(x^j/(1+x^j), j = 1 .. 80)/mul(1-x^j, j = 1 .. 80): gser := series(g, x = 0, 50): seq(coeff(gser, x, n), n = 0 .. 45); # _Emeric Deutsch_, Feb 08 2016
%t A209423 f[n_, i_] := Count[Flatten[IntegerPartitions[n]], i]
%t A209423 o[n_] := Sum[f[n, i], {i, 1, n, 2}]
%t A209423 e[n_] := Sum[f[n, i], {i, 2, n, 2}]
%t A209423 Table[o[n], {n, 1, 45}]  (* A066897 *)
%t A209423 Table[e[n], {n, 1, 45}]  (* A066898 *)
%t A209423 %% - %                   (* A209423 *)
%t A209423 b[n_, i_] := b[n, i] = Module[{m, f, g}, m = Mod[i, 2]; If[n==0, {1, 0, 0}, If[i<1, {0, 0, 0}, f = b[n, i-1]; g = If[i>n, {0, 0, 0}, b[n-i, i]]; {f[[1]] + g[[1]], f[[2]] + g[[2]] + m*g[[1]], f[[3]] + g[[3]] + (1-m)* g[[1]]}]]]; a[n_] := b[n, n][[2]] - b[n, n][[3]]; Table[a[n], {n, 1, 50}] (* _Jean-François Alcover_, Nov 16 2015, after _Alois P. Heinz_ *)
%Y A209423 Cf. A066897, A066898, A000041, A240009, A264398.
%K A209423 nonn
%O A209423 1,3
%A A209423 _Clark Kimberling_, Mar 08 2012