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%I #6 May 21 2018 11:27:42
%S 0,1,4,6,16,20,36,42,80,90,140,154,240,260,364,390,576,612,828,874,
%T 1200,1260,1628,1702,2256,2350,2964,3078,3920,4060,4980,5146,6464,
%U 6666,8092,8330,10224,10508,12540,12870,15600,15990,18900,19350,23056,23580,27508,28106,33216,33908
%N Expansion of x * (d/dx) Product_{k>=0} 1/(1 - x^(2^k)).
%C Sum of all parts of all partitions of n into powers of 2.
%C Convolution of the sequences A018819 and A038712.
%H <a href="/index/Par#part">Index entries for sequences related to partitions</a>
%F G.f.: x * (d/dx) Product_{k>=0} (1 + x^(2^k))^(k+1).
%F G.f.: Sum_{i>=0} 2^i*x^(2^i)/(1 - x^(2^i)) * Product_{j>=0} 1/(1 - x^(2^j)).
%F a(n) = n*A018819(k).
%t nmax = 49; CoefficientList[Series[x D[Product[1/(1 - x^2^k), {k, 0, Floor[Log[nmax]/Log[2]] + 1}], x], {x, 0, nmax}], x]
%t nmax = 49; CoefficientList[Series[x D[Product[(1 + x^2^k)^(k + 1), {k, 0, Floor[Log[nmax]/Log[2]] + 1}], x], {x, 0, nmax}], x]
%t a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d Boole[d == 2^IntegerExponent[d, 2]], {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[n a[n], {n, 0, 49}]
%Y Cf. A000079, A000123, A018819, A038712, A066186, A281688, A304908.
%K nonn
%O 0,3
%A _Ilya Gutkovskiy_, May 20 2018
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