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%I #15 Sep 12 2023 05:26:19
%S 1,-2,1,-4,6,-2,8,-8,1,-12,12,-4,14,-16,6,-16,18,-2,20,-24,8,-24,24,
%T -8,31,-28,1,-32,30,-12,32,-32,12,-36,48,-4,38,-40,14,-48,42,-16,44,
%U -48,6,-48,48,-16,57,-62,18,-56,54,-2,72,-64,20,-60,60,-24,62,-64,8,-64,84,-24,68,-72,24,-96,72,-8,74,-76,31,-80
%N Expansion of q*(psi(-q)*psi(-q^3))^2 in powers of q where psi() is a Ramanujan theta function.
%C Ramanujan theta functions: f(q) := Prod_{k>=1} (1-(-q)^k) (see A121373), phi(q) := theta_3(q) := Sum_{k=-oo..oo} q^(k^2) (A000122), psi(q) := Sum_{k=0..oo} q^(k*(k+1)/2) (A010054), chi(q) := Prod_{k>=0} (1+q^(2k+1)) (A000700).
%H G. C. Greubel, <a href="/A121456/b121456.txt">Table of n, a(n) for n = 1..1000</a>
%H Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>.
%F Expansion of (eta(q)*eta(q^3)*eta(q^4)*eta(q^12))^2/(eta(q^2)*eta(q^6))^2 in powers of q.
%F Euler transform of period 12 sequence [ -2, 0, -4, -2, -2, 0, -2, -2, -4, 0, -2, -4, ...].
%F Multiplicative with a(2^e) = -(2^e) if e>0, a(3^e) = 1, a(p^e) = (p^(e+1)-1)/(p-1) if p>3.
%F a(3*n)=a(n), a(4n+2)=-2*a(2*n+1).
%F a(n) = (-1)^(n+1)*A111932(n).
%F Dirichlet g.f.: (1 - 5/2^s + 1/2^(2*s-2) ) * (1 - 1/3^(s-1)) * zeta(s-1) * zeta(s). - _Amiram Eldar_, Sep 12 2023
%t eta[q_]:= q^(1/24)*QPochhammer[q]; a[n_]:= SeriesCoefficient[(eta[q] *eta[q^3]*eta[q^4]*eta[q^12])^2/(eta[q^2]*eta[q^6])^2, {q, 0, n}]; Table[a[n], {n, 1, 50}] (* _G. C. Greubel_, Mar 07 2018 *)
%o (PARI) {a(n)=if(n<1, 0, -(-1)^n*sumdiv(n,d,(n/d%2)*d*(d%3>0)))}
%Y Cf. A000122, A000700, A010054, A121373.
%K sign,easy,mult
%O 1,2
%A _Michael Somos_, Jul 30 2006
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