%I #21 May 10 2018 11:47:19
%S 1,0,1,-1,1,0,1,-1,1,-1,3,-2,3,-3,2,-1,4,-3,4,-4,7,-7,7,-7,9,-6,11,
%T -10,10,-11,15,-14,18,-19,21,-17,24,-22,26,-29,35,-34,42,-43,43,-39,
%U 52,-52,59,-59,74,-72,79,-87,93,-87,107,-108,118,-126,149,-146
%N Expansion of Product_{k>=1} (1 + x^(2*k) - x^(3*k)).
%H Vaclav Kotesovec, <a href="/A275821/b275821.txt">Table of n, a(n) for n = 0..10000</a>
%H Vaclav Kotesovec, <a href="/A275821/a275821.jpg">Graph - The asymptotic ratio (100000 terms)</a>
%F a(n) ~ (-1)^n * c^(1/4) * exp(sqrt(c*n)) / (2^(3/2)*sqrt(Pi)*n^(3/4)), where c = Integral_{0..infinity} log(1 + 2*exp(-x) + exp(-2*x) - exp(-3*x)) dx = 1.522848148277623680909526566...
%t nmax=100; CoefficientList[Series[Product[1+x^(2*k)-x^(3*k), {k, 1, nmax}], {x, 0, nmax}], x]
%t RootReduce[QPochhammer[Root[-1 + # + #^3 &, 1], x] QPochhammer[Root[-1 + # + #^3 &, 2], x] QPochhammer[Root[-1 + # + #^3 &, 3], x] + O[x]^70][[3]] (* _Vladimir Reshetnikov_, Nov 20 2016 *)
%t nmax = 100; p = ConstantArray[0, nmax + 1]; p[[1]] = 1; p[[3]] = 1; p[[4]] = -1; Do[Do[p[[j + 1]] = p[[j + 1]] + If[j < 2 k, 0, p[[j - 2 k + 1]]] - If[j < 3 k, 0, p[[j - 3 k + 1]]], {j, nmax, k, -1}];, {k, 2, nmax}]; p (* _Vaclav Kotesovec_, May 06 2018 *)
%Y Cf. A000726, A263401, A264905, A266686, A275820.
%K sign
%O 0,11
%A _Vaclav Kotesovec_, Nov 15 2016