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%I #11 Jul 09 2019 17:06:13
%S 1,4,8,16,24,32,48,64,88,96,128,128,176,192,256,256,344,352,448,384,
%T 512,512,640,512,688,704,896,768,1024,1024,1280,1024,1368,1376,1728,
%U 1408,1856,1792,2176,1536,2048,2048,2560,2048,2688,2560,3072,2048,2736,2752,3456
%N Expansion of Product_{k>=0} (1 + 2*x^(2^k))^2.
%C Self-convolution of A001316.
%F a(n) = Sum_{k=0..n} 2^(A000120(k)+A000120(n-k)).
%F a(n) = A001316(n) * Sum_{k=0..n} 2^(A007814(binomial(n,k))).
%F G.f. A(x) satisfies: A(x) = (1 + 2*x)^2 * A(x^2). - _Ilya Gutkovskiy_, Jul 09 2019
%t nmax = 50; CoefficientList[Series[Product[(1 + 2 x^(2^k))^2, {k, 0, Floor[Log[2, nmax]] + 1}], {x, 0, nmax}], x]
%t a[n_] := a[n] = Sum[2^(DigitCount[k, 2, 1] + DigitCount[n - k, 2, 1]), {k, 0, n}]; Table[a[n], {n, 0, 50}]
%Y Cf. A000120, A001316, A006046, A007814, A308986.
%K nonn
%O 0,2
%A _Ilya Gutkovskiy_, Jul 04 2019
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