A329801 Expansion of Sum_{k>=1} x^(k*(k + 1)/2) / (1 + x^(k*(k + 1)/2)).

1, -1, 2, -1, 1, -1, 1, -1, 2, 0, 1, -3, 1, -1, 3, -1, 1, -1, 1, -2, 3, -1, 1, -3, 1, -1, 2, 0, 1, -1, 1, -1, 2, -1, 1, -2, 1, -1, 2, -2, 1, -2, 1, -1, 4, -1, 1, -3, 1, 0, 2, -1, 1, -1, 2, -2, 2, -1, 1, -5, 1, -1, 3, -1, 1, 0, 1, -1, 2, 0, 1, -4, 1, -1, 3, -1, 1, 0, 1, -2, 2, -1, 1, -3, 1

Offset

1,3

Links

Table of n, a(n) for n=1..85.

Formula

G.f.: Sum_{k>=1} (-1)^(k + 1) * theta_2(x^(k/2)) / (2 * x^(k/8)).

a(n) = Sum_{d|n} (-1)^(n/d + 1) * A010054(d).

Mathematica

nmax = 85; CoefficientList[Series[Sum[x^(k (k + 1)/2)/(1 + x^(k (k + 1)/2)), {k, 1, nmax}], {x, 0, nmax}], x] // Rest
Table[Sum[(-1)^(n/d + 1) Boole[IntegerQ[Sqrt[8 d + 1]]], {d, Divisors[n]}], {n, 1, 85}]

Crossrefs

Cf. A000217, A007862, A010054, A048272, A304876, A317529.

Sequence in context: A087436 A340831 A334862 * A053255 A085856 A132126

Adjacent sequences: A329798 A329799 A329800 * A329802 A329803 A329804

Keyword

sign

Author

Ilya Gutkovskiy, Nov 21 2019

Status

approved