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%I #16 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,-2,0,
%U 6,2,0,0,-1,4,0,0,-2,0,0,0,-1,2,-2,-9,0,0,6,0
%N Expansion of (eta(q^2)^7 * eta(q^3)^2 * eta(q^12) / (eta(q)^2 * eta(q^4)^3 * eta(q^6)^3) - 1) / 2 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="/A138952/b138952.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 (phi(q) * phi(-q^2) * chi(-q^3) / chi(q^3) - 1) / 2 in powers of q where phi(), chi() are Ramanujan theta functions.
%F Moebius transform is period 24 sequence [1, -2, -4, 0, 1, 8, -1, 0, 4, -2, -1, 0, 1, 2, -4, 0, 1, -8, -1, 0, 4, 2, -1, 0, ...].
%F a(n) is multiplicative with a(2^e) = -1 if e>0, a(3^e) = -1 + 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 a(12*n + 7) = a(12*n + 11) = 0.
%F a(n) = -(-1)^n * A138950(n). 2 * a(n) = A138951(n).
%F a(2*n) = - A138950(n). a(2*n + 1) = A116604(n). - _Michael Somos_, Sep 07 2015
%F a(3*n + 1) = A258277(n). a(3*n + 2) = - A258278(n). - _Michael Somos_, Sep 07 2015
%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, KroneckerSymbol[ -4, n/#] {1, 1, -2}[[Mod[#, 3, 1]]] &]]; (* _Michael Somos_, Sep 07 2015 *)
%t a[ n_] := If[ n < 1, 0, Times @@ (Which[ # == 1, 1, # == 2, -1, # == 3, -1 + 2 (-1)^#2, Mod[#, 12] < 6, #2 + 1, True, 1 - Mod[#2, 2]] & @@@ FactorInteger@n)]; (* _Michael Somos_, Sep 07 2015 *)
%t a[ n_] := SeriesCoefficient[ (EllipticTheta[ 3, 0, q] EllipticTheta[ 4, 0, q^2] QPochhammer[ q^3] / QPochhammer[ -q^3] - 1) / 2, {q, 0, n}]; (* _Michael Somos_, Sep 07 2015 *)
%o (PARI) {a(n) = if( n<1, 0, -(-1)^n * sumdiv(n, d, kronecker(-4, n/d) * [-2, 1, 1][d%3 + 1]))};
%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, -1 + 2 * (-1)^e, p%12 < 6, e+1, 1-e%2 )))};
%Y Cf. A116604, A138950, A138951, A258277, A258278.
%K sign,mult
%O 1,3
%A _Michael Somos_, Apr 03 2008
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