A338045 G.f.: Sum_{k>=0} x^(2^k) / (1 - x^(2^k))^3.

1, 4, 6, 14, 15, 27, 28, 50, 45, 70, 66, 105, 91, 133, 120, 186, 153, 216, 190, 280, 231, 319, 276, 405, 325, 442, 378, 539, 435, 585, 496, 714, 561, 748, 630, 882, 703, 931, 780, 1100, 861, 1134, 946, 1309, 1035, 1357, 1128, 1581, 1225, 1600, 1326, 1820, 1431, 1863, 1540

Offset: 1

Author

Ilya Gutkovskiy, Oct 08 2020

Keywords

• nonn

Crossrefs

Cf. A000217, A000294, A000384, A001511, A014105, A129527, A195094, A209229, A328407, A338046.

Programs

Mathematica

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
a[n_] := If[EvenQ[n], a[n/2] + n (n + 1)/2, n (n + 1)/2]; Table[a[n], {n, 1, 55}]
Table[(1/2) DivisorSum[n, Boole[IntegerQ[Log[2, n/#]]] # (# + 1) &], {n, 1, 55}]

Formula

G.f. A(x) satisfies: A(x) = A(x^2) + x / (1 - x)^3.

a(2*n) = a(n) + A014105(n), a(2*n+1) = A000384(n+1).

a(n) = (1/2) * Sum_{d|n} A209229(n/d) * d * (d + 1).

a(n) = Sum_{d|n} A195094(d).

Product_{n>=1} (1 + x^n)^a(n) = g.f. for A000294.