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A208477 Difference between the sum of odd parts and the sum of even parts in all the partitions of n.

%I A208477 #23 Aug 29 2016 09:27:11
%S A208477 0,1,0,5,0,11,6,25,12,50,40,96,80,173,170,320,316,545,590,930,1020,
%T A208477 1552,1760,2537,2900,4066,4736,6450,7540,10045,11856,15482,18280,
%U A208477 23555,27920,35461,42032,52805,62662,77955,92380,113963,135040,165295,195540,237866
%N A208477 Difference between the sum of odd parts and the sum of even parts in all the partitions of n.
%H A208477 Alois P. Heinz, <a href="/A208477/b208477.txt">Table of n, a(n) for n = 0..1000</a>
%F A208477 a(n) = A066967(n) - A066966(n).
%F A208477 G.f.: (Sum_{i>0} (2*i-1)*x^(2*i-1)/(1-x^(2*i-1))-2*i*x^(2*i)/(1-x^(2*i))) / Product_{j>0} (1-x^j). - _Alois P. Heinz_, Mar 10 2012
%p A208477 b:= proc(n,i) option remember; local g, h;
%p A208477       if n=0 then [1, 0]
%p A208477     elif i<1 then [0, 0]
%p A208477     else g:= b(n, i-1);
%p A208477          h:= `if`(i>n, [0, 0], b(n-i, i));
%p A208477          [g[1]+h[1], g[2]+h[2] +h[1]*i*(2*(i mod 2)-1)]
%p A208477       fi
%p A208477     end:
%p A208477 a:= n-> b(n, n)[2]:
%p A208477 seq(a(n), n=0..60); # _Alois P. Heinz_, Mar 10 2012
%t A208477 Map[Total[Select[#, OddQ]] - Total[Select[#, EvenQ]] &[Flatten[IntegerPartitions[#]]] &, -1 + Range[30]] (* _Peter J. C. Moses_, Mar 14 2014 *)
%t A208477 max = 60; s = Sum[x^(2i) (x^(2i) - 2i (x-1) - 1)/(x + x^(4i) - (x+1) x^(2i) ), {i, 1, Floor[max/2]}]/QPochhammer[x] + O[x]^max; CoefficientList[s, x] (* _Jean-François Alcover_, Aug 29 2016, after _Alois P. Heinz_ *)
%Y A208477 Cf. A066186, A209423.
%K A208477 nonn
%O A208477 0,4
%A A208477 _Omar E. Pol_, Mar 10 2012
%E A208477 More terms from _Alois P. Heinz_, Mar 10 2012