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%I A022693 #31 Sep 08 2022 08:44:46
%S A022693 1,-1,-1,-2,2,-1,4,-1,18,-22,12,-26,67,-86,42,-235,432,-364,506,-868,
%T A022693 1434,-2396,2225,-3348,10842,-11822,8049,-24468,36662,-40024,69766,
%U A022693 -96052,171976,-278242,251886,-419723,885806,-998468,1103660,-2381042,4009539,-4478416,6372514,-9913690
%N A022693 Expansion of Product_{m>=1} 1/(1 + m*q^m).
%H A022693 Vaclav Kotesovec, <a href="/A022693/b022693.txt">Table of n, a(n) for n = 0..5000</a>
%F A022693 From _Vaclav Kotesovec_, Dec 15 2015: (Start)
%F A022693 a(n) ~ (-1)^n * c * 3^(n/3), where
%F A022693 c = 2.0319526534291644237634198503666896166412... if mod(n,3) = 0
%F A022693 c = 1.8420902462379331740718256785549611496880... if mod(n,3) = 1
%F A022693 c = 1.6677871810486313099783673373643842640151... if mod(n,3) = 2.
%F A022693 (End)
%F A022693 From _Benedict W. J. Irwin_, Mar 19 2017: (Start)
%F A022693 Conjecture: a(n) = Sum_{i_1,i_2,i_3,...}[(-1)^(i_1+i_2+i_3+...)*Product_{n>0} n^i_n], where the sum is over all valid sequences of positive i_k such that i_1+2*i_2+3*i_3+4*i_4+...= n.
%F A022693 Examples: Setting i_k=0 unless explicitly mentioned.
%F A022693   n=1, (i_1=1), a(1)= -1^1 = -1.
%F A022693   n=2, (i_1=2) or (i_2=1), a(2) = 1^2 - 2^1 = -1.
%F A022693   n=3, (i_1=3) or (i_1=1,i_2=1) or (i_3=1), a(3)=-1^3 + 1^1*2^1 - 3^1 = -2.
%F A022693 (End)
%t A022693 nmax = 40; CoefficientList[Series[Product[1/(1 + k*x^k), {k, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Dec 15 2015 *)
%t A022693 nmax = 40; CoefficientList[Series[Exp[-Sum[(-1)^(j+1)*PolyLog[-j, x^j]/j, {j, 1, nmax}]], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Dec 15 2015 *)
%o A022693 (PARI) m=50; q='q+O('q^m); Vec(prod(n=1,m,1/(1+n*q^n))) \\ _G. C. Greubel_, Feb 25 2018
%o A022693 (Magma) Coefficients(&*[1/(1+m*x^m):m in [1..40]])[1..40] where x is PolynomialRing(Integers()).1; // _G. C. Greubel_, Feb 25 2018
%Y A022693 Cf. A006906, A022629, A022661, A266971.
%K A022693 sign
%O A022693 0,4
%A A022693 _N. J. A. Sloane_

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