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%I #45 Sep 08 2022 08:44:50
%S 1,-1,-1,-1,1,2,-1,2,0,0,-1,-1,-1,-1,0,1,-1,-1,2,0,1,2,1,-1,0,-1,2,-1,
%T 0,-1,-1,0,-1,-1,0,-1,-2,2,2,0,-1,1,0,1,0,-1,2,2,1,0,-2,2,-1,0,-1,-1,
%U -1,1,-1,0,0,-1,-1,-1,0,0,2,-2,-1,0,-1,1,2,2,0,0
%N Expansion of q^(-1/6) * eta(q) * eta(q^3) in powers of q.
%C Number 65 of the 74 eta-quotients listed in Table I of Martin (1996).
%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%C Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).
%H Seiichi Manyama, <a href="/A030203/b030203.txt">Table of n, a(n) for n = 0..10000</a>
%H S. R. Finch, <a href="http://arXiv.org/abs/math.NT/0701251">Powers of Euler's q-Series</a>, (arXiv:math.NT/0701251).
%H M. Koike, <a href="http://projecteuclid.org/euclid.nmj/1118787564">On McKay's conjecture</a>, Nagoya Math. J., 95 (1984), 85-89.
%H Y. Martin, <a href="http://dx.doi.org/10.1090/S0002-9947-96-01743-6">Multiplicative eta-quotients</a>, Trans. Amer. Math. Soc. 348 (1996), no. 12, 4825-4856, see page 4852 Table I.
%H Michael Somos, <a href="/A030203/a030203.txt">Index to Yves Martin's list of 74 multiplicative eta-quotients and their A-numbers</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 f(-x) * f(-x^3) where f(-x) := f(-x, -x^2) is a Ramanujan theta function. - _Michael Somos_, Jul 27 2006
%F Expansion of q^(-1/6) * sqrt(b(q) * c(q)/3) in powers of q where b(), c() are cubic AGM theta functions. - _Michael Somos_, Nov 01 2006
%F Euler transform of period 3 sequence [ -1, -1, -2, ...]. - _Michael Somos_, Jul 27 2006
%F Given g.f. A(x), then B(q) = (q * A(q^6))^2 satisfies 0 = f(B(q), B(q^2), B(q^4)) where f(u, v, w) = v^3 - u^2*w - 4*u*w^2. - _Michael Somos_, Jul 27 2006
%F a(n) = b(6*n + 1) where b(n) is multiplicative and b(2^e) = b(3^e) = 0^e, b(p^e) = (1+(-1)^e)/2 if p == 5 (mod 6), b(p^e) = e+1 if p = x^2 + 27*y^2, b(p^e) = [1, -1, 0] depending on e (mod 3) otherwise.
%F G.f. is a period 1 Fourier series which satisfies f(-1 / (108 t)) = 108^(1/2) (t/i) f(t) where q = exp(2 Pi i t). - _Michael Somos_, Jan 22 2012
%F G.f.: Product_{k>0} (1 - x^k) * (1 - x^(3*k)).
%e G.f. = 1 - x - x^2 - x^3 + x^4 + 2*x^5 - x^6 + 2*x^7 - x^10 - x^11 - x^12 - x^13 + ...
%e G.f. = q - q^7 - q^13 - q^19 + q^25 + 2*q^31 - q^37 + 2*q^43 - q^61 - q^67 - ...
%t a[ n_] := SeriesCoefficient[ QPochhammer[ x] QPochhammer[ x^3], {x, 0, n}]; (* _Michael Somos_, Jan 31 2015 *)
%o (PARI) {a(n) = my(A, p, e); if( n<0, 0, n = 6*n + 1; A = factor(n); prod( k=1, matsize(A)[1], [p, e] = A[k,]; if( p<5, 0, p%6==5, (1 + (-1)^e) / 2, (p-1) / znorder( Mod(2, p))%3, kronecker( e+1, 3), e+1)))}; /* _Michael Somos_, Jul 27 2006 */
%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^3 + A), n))}; /* _Michael Somos_, Jul 27 2006 */
%o (Magma) Basis( CuspForms( Gamma1(108), 1), 452)[1]; /* _Michael Somos_, Jan 31 2015 */
%K sign
%O 0,6
%A _N. J. A. Sloane_.
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