%I #11 Dec 30 2023 12:36:59

%S 1,-1,-3,-2,1,10,13,15,-1,-30,-63,-89,-80,-14,131,304,493,561,434,-32,

%T -836,-1895,-2960,-3583,-3240,-1338,2401,8004,14499,20494,23369,20401,

%U 8567,-13741,-46408,-85717,-124027,-149612,-147167,-101002,2520,168026,388077,634914

%N Expansion of Product_{k>=1} (1 - x^k)^(2*k-1).

%F G.f.: exp(Sum_{k>=1} (sigma_1(k) - 2*sigma_2(k))*x^k/k).

%p a:=series(mul((1-x^k)^(2*k-1),k=1..100),x=0,44): seq(coeff(a,x,n),n=0..43); # _Paolo P. Lava_, Apr 02 2019

%t nmax = 43; CoefficientList[Series[Product[(1 - x^k)^(2 k - 1), {k, 1, nmax}], {x, 0, nmax}], x]

%t nmax = 43; CoefficientList[Series[Exp[Sum[(DivisorSigma[1, k] - 2 DivisorSigma[2, k]) x^k/k, {k, 1, nmax}]], {x, 0, nmax}], x]

%t a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d (1 - 2 d), {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 43}]

%Y Cf. A000203, A001157, A073592, A253289, A255835, A276551, A278945, A285069.

%K sign

%O 0,3

%A _Ilya Gutkovskiy_, Sep 25 2018