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%I #31 Jan 03 2020 16:16:46
%S -1,0,-3,4,-5,6,-7,16,-18,20,-22,36,-39,42,-60,80,-85,90,-114,140,
%T -168,176,-207,264,-300,312,-378,448,-493,540,-620,736,-825,884,-1015,
%U 1188,-1295,1406,-1599,1840,-2009,2184,-2451,2772,-3060,3312,-3666,4176,-4557
%N Sum of all parts of all partitions of n into an even number of parts minus the sum of all parts of all partitions of n into an odd number of parts.
%C Zero together with the absolute values of this sequence give A330373. - _Omar E. Pol_, Dec 31 2019
%H Clark Kimberling, <a href="/A235324/b235324.txt">Table of n, a(n) for n = 1..500</a>
%F a(n) = n*A081362(n) for n >= 1.
%F G.f.: x*f'(x), where f(x) = Product_{k > 0} (1 - x^(2*k - 1)).
%F a(n) ~ (-1)^n * exp(Pi*sqrt(n/6)) * n^(1/4) / (2^(7/4) * 3^(1/4)). - _Vaclav Kotesovec_, Nov 21 2016
%e There are 12 partitions of 8 into an even number of parts and 10 into an odd number of parts, so that a(8) = 12*8 - 10*8 = 16.
%t Table[n*SeriesCoefficient[Product[1-x^k, {k, 1, n, 2}], {x,0,n}], {n,70}]
%t nmax = 60; Rest[CoefficientList[Series[x*D[Product[1/(1 + x^k), {k, 1, nmax}], x], {x, 0, nmax}], x]] (* _Vaclav Kotesovec_, Nov 21 2016 *)
%o (PARI) a(n)= my(A); n*if(n<0, 0, A= x*O(x^n); polcoeff(eta(x+A)/eta(x^2+A), n));
%o vector(50, n, a(n) ) /* from Michael Somos code in A081362 */ \\ _G. C. Greubel_, Dec 31 2019
%Y Cf. A000041, A027187, A027193, A081362, A330373.
%K easy,sign
%O 1,3
%A _Clark Kimberling_, Mar 27 2014
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