A372625 Expansion of Sum_{k>=1} k^2 * x^(k^2) / (1 + x^k).

1, -1, 1, 3, 1, -5, 1, 3, 10, -5, 1, -6, 1, -5, 10, 19, 1, -14, 1, -13, 10, -5, 1, 10, 26, -5, 10, -13, 1, -39, 1, 19, 10, -5, 26, 14, 1, -5, 10, -6, 1, -50, 1, -13, 35, -5, 1, 46, 50, -30, 10, -13, 1, -50, 26, -30, 10, -5, 1, -11, 1, -5, 59, 83, 26, -50, 1, -13, 10, -79

Offset: 1

Author

Ilya Gutkovskiy, May 07 2024

Keywords

• sign

Crossrefs

Cf. A064027, A066839, A078306, A095118, A321543, A321558, A348608.

Programs

Mathematica

nmax = 70; CoefficientList[Series[Sum[k^2 x^(k^2)/(1 + x^k), {k, 1, nmax}], {x, 0, nmax}], x] // Rest
Table[DivisorSum[n, (-1)^(# + n/#) #^2 &, # <= Sqrt[n] &], {n, 1, 70}]

Formula

a(n) = Sum_{d|n, d <= sqrt(n)} (-1)^(d + n/d) * d^2.