Reminder: The OEIS is hiring a new managing editor, and the application deadline is January 26.
%I #13 Mar 12 2021 22:24:45
%S 1,-1,3,-1,2,-3,0,-1,1,-2,0,-3,2,0,6,-1,2,-1,0,-2,0,0,0,-3,3,-2,3,0,2,
%T -6,0,-1,0,-2,0,-1,2,0,6,-2,2,0,0,0,2,0,0,-3,1,-3,6,-2,2,-3,0,0,0,-2,
%U 0,-6,2,0,0,-1,4,0,0,-2,0,0,0,-1,2,-2,9,0,0,-6
%N Expansion of 1 - eta(q) * eta(q^3) * eta(q^4)^3 / (eta(q^2)^2 * eta(q^12)) in powers of q.
%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H G. C. Greubel, <a href="/A138746/b138746.txt">Table of n, a(n) for n = 1..10000</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 1 - psi(-q) * psi(q^2) * chi(-q^3) * chi(-q^6) in powers of q where psi(), chi() are Ramanujan theta functions.
%F Moebius transform is period 24 sequence [ 1, -2, 2, 0, 1, -4, -1, 0, -2, -2, -1, 0, 1, 2, 2, 0, 1, 4, -1, 0, -2, 2, -1, 0, ...].
%F a(n) is multiplicative with a(2^e) = -1 if e>0, a(3^e) = 2 - (-1)^e, a(p^e) = e+1 if p == 1, 5 (mod 12), a(p^e) = (1 + (-1)^e) / 2 if p == 7, 11 (mod 12).
%F G.f.: Sum_{k>0} -(-1)^k * ( f(6*k - 1) + 2 * f(6*k - 3) + f(6*k - 5) ) where f(k) := x^k / (1 + x^k).
%F a(12*n + 7) = a(12*n + 11) = 0.
%F a(n) = - A138745(n) unless n=0.
%e G.f = q - q^2 + 3*q^3 - q^4 + 2*q^5 - 3*q^6 - q^8 + q^9 - 2*q^10 - 3*q^12 + ...
%t a[ n_] := If[ n < 1, 0, -(-1)^n DivisorSum[ n, (-1)^Quotient[#, 6] {1, 0, 2, 0, 1, 0}[[Mod[#, 6, 1]]] &]]; (* _Michael Somos_, Sep 08 2015 *)
%t a[ n_] := If[ n < 1, 0, Times @@ (Which[ # < 3, -(-1)^#, # == 3, Mod[#2, 2] 2 + 1, Mod[#, 4] == 1, #2 + 1, True, 1 - Mod[#2, 2]] & @@@ FactorInteger @ n)]; (* _Michael Somos_, Sep 08 2015 *)
%t QP = QPochhammer; s = (1/q)*(1-QP[q]*QP[q^3]*(QP[q^4]^3/(QP[q^2]^2*QP[q^12] ))) + O[q]^80; CoefficientList[s, q] (* _Jean-François Alcover_, Nov 27 2015 *)
%o (PARI) {a(n) = if( n<1, 0, -(-1)^n * sumdiv(n, d, ((d%2) * ((d%3==0) + 1)) * (-1)^(d\6)))};
%o (PARI) {a(n) = my(A, p, e); if( n<1, 0, A = factor(n); prod(k=1, matsize(A)[1], [p, e] = A[k, ]; if( p==2, -1, p==3, 2 - (-1)^e, p%12<6, e+1, 1-e%2)))};
%Y Cf. A138745.
%K sign,mult
%O 1,3
%A _Michael Somos_, Mar 27 2008