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A194796 Imbalance of the number of parts of all partitions of n.

%I A194796 #21 Nov 11 2015 11:20:28
%S A194796 0,-1,0,-3,0,-8,0,-17,3,-31,10,-58,22,-101,52,-167,104,-278,191,-451,
%T A194796 344,-711,594,-1119,983,-1730,1606,-2635,2555,-3990,3978,-5972,6118,
%U A194796 -8835,9269,-12986,13835,-18917,20454,-27320,29900,-39204,43268,-55846,62112
%N A194796 Imbalance of the number of parts of all partitions of n.
%C A194796 Consider the three-dimensional structure of the shell model of partitions, version "tree" (see the illustration in A194795). Note that only the parts > 1 produce the imbalance. The 1's are located in the central columns therefore they do not produce the imbalance. For more information see A135010.
%H A194796 Alois P. Heinz, <a href="/A194796/b194796.txt">Table of n, a(n) for n = 1..1000</a>
%F A194796 a(n) = Sum_{k=1..n} (-1)^(k-1)*A138135(k).
%p A194796 b:= proc(n, i) option remember; local f, g;
%p A194796       if n=0 or i=1 then [1, 0]
%p A194796     else f:= b(n, i-1); g:= `if`(i>n, [0, 0], b(n-i, i));
%p A194796          [f[1]+g[1], f[2]+g[2]+g[1]]
%p A194796       fi
%p A194796     end:
%p A194796 a:= proc(n) option remember;
%p A194796       (-1)^n*(b(n-1, n-1)[2]-b(n, n)[2])+`if`(n=1, 0, a(n-1))
%p A194796     end:
%p A194796 seq(a(n), n=1..60);  # _Alois P. Heinz_, Apr 04 2012
%t A194796 b[n_, i_] := b[n, i] = Module[{f, g}, If[n == 0 || i == 1, {1, 0}, f = b[n, i-1]; g = If[i>n, {0, 0}, b[n-i, i]]; {f[[1]] + g[[1]], f[[2]] + g[[2]] + g[[1]]}]]; a[n_] := a[n] = (-1)^n*(b[n-1, n-1][[2]] - b[n, n][[2]]) + If[n == 1, 0, a[n-1]]; Table [a[n], {n, 1, 60}] (* _Jean-François Alcover_, Nov 11 2015, after _Alois P. Heinz_ *)
%o A194796 (PARI) vector(50, n, sum(k=1, n, (-1)^(k-1)*(numdiv(k)-1+sum(j=1, k-1, (numdiv(j)-1)*(numbpart(k-j)-numbpart(k-j-1)))))) \\ _Altug Alkan_, Nov 11 2015
%Y A194796 Cf. A006128, A096541, A135010, A138121, A138135, A138137, A197595, A194797, A194809.
%K A194796 sign
%O A194796 1,4
%A A194796 _Omar E. Pol_, Feb 01 2012
%E A194796 More terms from _Alois P. Heinz_, Apr 04 2012