login
a(n) = -(1/n) * Sum_{d|n} phi(n/d) * (-4)^d.
2

%I #7 Apr 16 2021 15:42:06

%S 4,-6,24,-66,208,-676,2344,-8226,29144,-104760,381304,-1398476,

%T 5162224,-19172796,71582944,-268439586,1010580544,-3817734596,

%U 14467258264,-54975633768,209430787824,-799644629556,3059510616424,-11728124734476,45035996273872,-173215367702376,667199944815064

%N a(n) = -(1/n) * Sum_{d|n} phi(n/d) * (-4)^d.

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

%F a(n) = -(1/n) * Sum_{k=1..n} (-4)^gcd(n,k).

%F Product_{n>=1} 1 / (1 - x^n)^a(n) = g.f. for A261568.

%t Table[-(1/n) Sum[EulerPhi[n/d] (-4)^d, {d, Divisors[n]}], {n, 1, 27}]

%t nmax = 27; CoefficientList[Series[Sum[EulerPhi[k] Log[1 + 4 x^k]/k, {k, 1, nmax}], {x, 0, nmax}], x] // Rest

%Y Cf. A000010, A001868, A038065, A074763, A261568, A343465, A343467.

%K sign

%O 1,1

%A _Ilya Gutkovskiy_, Apr 16 2021