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A208978 Expansion of f(x) * f(x^3) where f() is a Ramanujan theta function.

%I A208978 #25 Feb 16 2025 08:33:16
%S A208978 1,1,-1,1,1,-2,-1,-2,0,0,-1,1,-1,1,0,-1,-1,1,2,0,1,-2,1,1,0,1,2,1,0,1,
%T A208978 -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,-1,
%U A208978 -1,-1,0,0,1,-1,1,0,0,2,2,-1,0,-1,-1,2,-2,0,0,2
%N A208978 Expansion of f(x) * f(x^3) where f() is a Ramanujan theta function.
%C A208978 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%C A208978 Number 72 of the 74 eta-quotients listed in Table I of Martin (1996).
%H A208978 Seiichi Manyama, <a href="/A208978/b208978.txt">Table of n, a(n) for n = 0..10000</a>
%H A208978 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 A208978 Michael Somos, <a href="/A030203/a030203.txt">Index to Yves Martin's list of 74 multiplicative eta-quotients and their A-numbers</a>
%H A208978 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A208978 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A208978 Expansion of q^(-1/6) * (eta(q^2) * eta(q^6))^3 / (eta(q) * eta(q^3) * eta(q^4) * eta(q^12)) in powers of q.
%F A208978 Euler transform of period 12 sequence [ 1, -2, 2, -1, 1, -4, 1, -1, 2, -2, 1, -2, ...].
%F A208978 G.f. is a period 1 Fourier series which satisfies f(-1 / (432 t)) = 432^(1/2) (t/i) f(t) where q = exp(2 Pi i t).
%F A208978 a(n) = b(6*n + 1) where b(n) is multiplicative with b(2^e) = b(3^e) = 0^e, b(p^e) = (1 + (-1)^e) / 2 if p == 5 (mod 6), b(p^e) = (-1)^(floor(p/6) * e) * t if p == 1 (mod 6) where t = e+1 if p = x^2 + 27*y^2, and t = [1, -1, 0] depending on e (mod 3) otherwise.
%F A208978 G.f.: Product_{k>0} (1 - (-x)^k) * (1 - (-x)^(3*k)).
%F A208978 a(n) = (-1)^n * A030203(n).
%e A208978 G.f. = 1 + x - x^2 + x^3 + x^4 - 2*x^5 - x^6 - 2*x^7 - x^10 + x^11 - x^12 + ...
%e A208978 G.f. = q + q^7 - q^13 + q^19 + q^25 - 2*q^31 - q^37 - 2*q^43 - q^61 + q^67 + ...
%t A208978 a[ n_] := SeriesCoefficient[ QPochhammer[ -x] QPochhammer[ -x^3] , {x, 0, n}]; (* _Michael Somos_, Jun 09 2015 *)
%o A208978 (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, (-1)^(p\6*e) * if( (p-1) / znorder( Mod(2, p))%3, kronecker( e+1, 3), e+1))))};
%o A208978 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A) * eta(x^6 + A))^3 / (eta(x + A) * eta(x^3 + A) * eta(x^4 + A) * eta(x^12 + A)), n))};
%Y A208978 Cf. A030203.
%K A208978 sign
%O A208978 0,6
%A A208978 _Michael Somos_, Mar 03 2012