A317529 Expansion of Sum_{k>=1} x^(k^2)/(1 + x^(k^2)).

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

Offset: 1

Author

Ilya Gutkovskiy, Jul 30 2018

Keywords

• sign
• mult
• easy

Links

• Antti Karttunen, Table of n, a(n) for n = 1..11025
• Antti Karttunen, Data supplement: n, a(n) computed for n = 1..100000

Crossrefs

Cf. A000290, A010052, A046951, A048272, A113024, A300853.

Programs

Maple

seq(coeff(series(add(x^(k^2)/(1+x^(k^2)),k=1..n), x,n+1),x,n),n=1..100); # Muniru A Asiru, Jul 30 2018

Mathematica

nmax = 95; Rest[CoefficientList[Series[Sum[x^k^2/(1 + x^k^2), {k, 1, nmax}], {x, 0, nmax}], x]]
nmax = 95; Rest[CoefficientList[Series[Log[Product[(1 + x^k^2)^(1/k^2), {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]]
Table[DivisorSum[n, (-1)^(n/# + 1) &, IntegerQ[#^(1/2)] &], {n, 95}]
f[p_, e_] := If[p == 2, If[OddQ[e], -Floor[e/2 + 1], -Floor[(e - 1)/2]], Floor[e/2 + 1]]; a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Oct 25 2020 *)

PARI

A317529(n) = sumdiv(n,d,((-1)^(1+(n/d)))*issquare(d)); \\ Antti Karttunen, Nov 07 2018

Formula

G.f.: Sum_{k>=1} x^A000290(k)/(1 + x^A000290(k)).

L.g.f.: log(Product_{k>=1} (1 + x^(k^2))^(1/k^2)) = Sum_{n>=1} a(n)*x^n/n.

a(n) = Sum_{d|n} (-1)^(n/d+1)*A010052(d).

If n is odd, a(n) = A046951(n).

Multiplicative with a(2^e) = -floor(e/2+1) for odd e, -floor((e-1)/2) for even e, and a(p^e) = floor(e/2+1) for an odd prime p. - Amiram Eldar, Oct 25 2020

From Amiram Eldar, Dec 18 2023: (Start)

Dirichlet g.f.: zeta(s) * zeta(2*s) * (1 - 1/2^(s-1)).

Sum_{k=1..n} a(k) ~ c * sqrt(n), where c = -(sqrt(2)-1) * zeta(1/2) = 0.604898... (A113024). (End)