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%I #21 Sep 09 2023 12:35:22
%S 1,-1,-2,-2,-3,-1,-2,6,12,36,74,162,301,599,1090,1986,3479,5993,9852,
%T 15644,23094,30690,31868,9068,-82372,-345308,-1010956,-2577868,
%U -6098822,-13751218,-29962588,-63604140,-132205949,-269982371,-542866266,-1076420666
%N Expansion of Product_{k>0} (1 - x^k)^(2^(k-1)) in powers of x.
%H Seiichi Manyama, <a href="/A200751/b200751.txt">Table of n, a(n) for n = 0..1000</a>
%F Let F(a, x) = (1 - a) * (1 - a*x)^2 * (1 - a*x^2)^4 * ... where |x|<1/2. Then F(a, x) = (1 - a) * F(a*x, x)^2 and g.f. A(x) = F(x, x).
%F Euler transform of [ -1, -2, -4, -8, -16, ... ].
%F G.f.: (1 - x) * (1 - x^2)^2 * (1 - x^3)^4 * ...
%F Convolution inverse of A034691.
%F a(0) = 1; a(n) = -(1/n) * Sum_{k=1..n} A083413(k) * a(n-k). - _Seiichi Manyama_, Jul 17 2023
%F a(n) = Sum_{k=0..2n} (-1)^k*A360634(2n,k). - _Alois P. Heinz_, Sep 09 2023
%e 1 - x - 2*x^2 - 2*x^3 - 3*x^4 - x^5 - 2*x^6 + 6*x^7 + 12*x^8 + 36*x^9 + ...
%t a[n_] := SeriesCoefficient[Series[Product[(1 - x^k)^2^(k - 1),
%t {k, n}], {x, 0, n}], n]; Table[a[n], {n, 0, 35}] (* _T. D. Noe_, Nov 23 2011 *)
%o (PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( prod( k=1, n, (1 - x^k + A) ^ 2^(k - 1)), n))}
%Y Cf. A034691, A083413, A360634.
%K sign
%O 0,3
%A _Michael Somos_, Nov 21 2011
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