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%I #5 Oct 09 2020 05:18:14
%S 1,4,6,14,15,27,28,50,45,70,66,105,91,133,120,186,153,216,190,280,231,
%T 319,276,405,325,442,378,539,435,585,496,714,561,748,630,882,703,931,
%U 780,1100,861,1134,946,1309,1035,1357,1128,1581,1225,1600,1326,1820,1431,1863,1540
%N G.f.: Sum_{k>=0} x^(2^k) / (1 - x^(2^k))^3.
%F G.f. A(x) satisfies: A(x) = A(x^2) + x / (1 - x)^3.
%F a(2*n) = a(n) + A014105(n), a(2*n+1) = A000384(n+1).
%F a(n) = (1/2) * Sum_{d|n} A209229(n/d) * d * (d + 1).
%F a(n) = Sum_{d|n} A195094(d).
%F Product_{n>=1} (1 + x^n)^a(n) = g.f. for A000294.
%t nmax = 55; CoefficientList[Series[Sum[x^(2^k) /(1 - x^(2^k))^3, {k, 0, Floor[Log[2, nmax]] + 1}], {x, 0, nmax}], x] // Rest
%t a[n_] := If[EvenQ[n], a[n/2] + n (n + 1)/2, n (n + 1)/2]; Table[a[n], {n, 1, 55}]
%t Table[(1/2) DivisorSum[n, Boole[IntegerQ[Log[2, n/#]]] # (# + 1) &], {n, 1, 55}]
%Y Cf. A000217, A000294, A000384, A001511, A014105, A129527, A195094, A209229, A328407, A338046.
%K nonn
%O 1,2
%A _Ilya Gutkovskiy_, Oct 08 2020
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