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%I #5 Jul 04 2019 15:22:59
%S 1,-2,2,-4,10,-20,36,-72,154,-308,596,-1192,2420,-4840,9608,-19216,
%T 38586,-77172,154036,-308072,616740,-1233480,2465768,-4931536,9865492,
%U -19730984,39457128,-78914256,157838120,-315676240,631333264,-1262666528,2525371642,-5050743284
%N Expansion of Product_{k>=0} 1/(1 + 2*x^(2^k)).
%C Convolution inverse of A001316.
%F a(0) = 1; a(n) = -Sum_{k=1..n} 2^A000120(k) * a(n-k).
%t nmax = 33; CoefficientList[Series[Product[1/(1 + 2 x^(2^k)), {k, 0, Floor[Log[2, nmax]] + 1}], {x, 0, nmax}], x]
%t a[0] = 1; a[n_] := a[n] = -Sum[2^DigitCount[k, 2, 1] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 33}]
%Y Cf. A000120, A001316, A006046, A308985.
%K sign
%O 0,2
%A _Ilya Gutkovskiy_, Jul 04 2019