Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%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