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A328408 G.f. A(x) satisfies: A(x) = A(x^2) + x * (1 + 4*x + x^2) / (1 - x)^4. 1

%I

%S 1,9,27,73,125,243,343,585,729,1125,1331,1971,2197,3087,3375,4681,

%T 4913,6561,6859,9125,9261,11979,12167,15795,15625,19773,19683,25039,

%U 24389,30375,29791,37449,35937,44217,42875,53217,50653,61731,59319,73125,68921,83349,79507,97163,91125

%N G.f. A(x) satisfies: A(x) = A(x^2) + x * (1 + 4*x + x^2) / (1 - x)^4.

%F G.f.: Sum_{k>=0} x^(2^k) * (1 + 4*x^(2^k) + x^(2^(k+1))) / (1 - x^(2^k))^4.

%F G.f.: (1/7) * Sum_{k>=1} J_3(2*k) * x^k / (1 - x^k), where J_3() is the Jordan function (A059376).

%F Dirichlet g.f.: zeta(s-3) / (1 - 2^(-s)).

%F a(2*n) = a(n) + 8*n^3, a(2*n+1) = (2*n + 1)^3.

%F a(n) = Sum_{d|n} A209229(n/d) * d^3.

%F Product_{n>=1} (1 + x^n)^a(n) = g.f. for A023872.

%F Sum_{k=1..n} a(k) ~ 4*n^4/15. - _Vaclav Kotesovec_, Oct 15 2019

%t nmax = 45; CoefficientList[Series[Sum[x^(2^k) (1 + 4 x^(2^k) + x^(2^(k + 1)))/(1 - x^(2^k))^4, {k, 0, Floor[Log[2, nmax]] + 1}], {x, 0, nmax}], x] // Rest

%t a[n_] := If[EvenQ[n], a[n/2] + n^3, n^3]; Table[a[n], {n, 1, 45}]

%t Table[DivisorSum[n, Boole[IntegerQ[Log[2, n/#]]] #^3 &], {n, 1, 45}]

%o (MAGMA) [n eq 1 select 1 else IsOdd(n) select n^3 else Self(n div 2)+n^3 :n in [1..45]]; // _Marius A. Burtea_, Oct 15 2019

%Y Cf. A000578, A001511, A016755, A023872, A059376, A129527, A209229, A328407.

%K nonn,mult

%O 1,2

%A _Ilya Gutkovskiy_, Oct 14 2019

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Last modified January 24 16:47 EST 2020. Contains 331209 sequences. (Running on oeis4.)