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A256014 Expansion of phi(-q^3)^4 / (phi(-q) * phi(-q^9)) in powers of q where phi() is a Ramanujan theta function.

%I A256014 #22 Feb 16 2025 08:33:25
%S A256014 1,2,4,0,-2,-8,0,0,4,-4,-4,0,0,4,0,0,-2,-8,4,0,8,0,0,0,0,6,8,0,0,-8,0,
%T A256014 0,4,0,-4,0,4,4,0,0,-4,-8,0,0,0,-8,0,0,0,2,12,0,-4,-8,0,0,0,0,-4,0,0,
%U A256014 4,0,0,-2,-16,0,0,8,0,0,0,4,4,8,0,0,0,0,0,8
%N A256014 Expansion of phi(-q^3)^4 / (phi(-q) * phi(-q^9)) in powers of q where phi() is a Ramanujan theta function.
%C A256014 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A256014 G. C. Greubel, <a href="/A256014/b256014.txt">Table of n, a(n) for n = 0..1000</a>
%H A256014 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A256014 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A256014 Expansion of eta(q^2) * eta(q^3)^8 * eta(q^18) / (eta(q)^2 * eta(q^6)^4 * eta(q^9)^2) in powers of q.
%F A256014 Euler transform of period 18 sequence [ 2, 1, -6, 1, 2, -3, 2, 1, -4, 1, 2, -3, 2, 1, -6, 1, 2, -2, ...].
%F A256014 a(n) = (-1)^n * A256280(n). a(3*n + 1) = 2 * A258277(n). a(3*n + 2) = 4 * A258278(n). a(4*n) = A256280(n). a(4*n + 3) = a(9*n + 3) = a(9*n + 6) = 0.
%F A256014 a(6*n + 2) = 4 * A122865(n). a(6*n + 4) = -2 * A122856(n). a(9*n) = A104794(n). a(12*n + 1) = A002175(n). a(12*n + 5) = -8 * A121444(n).
%e A256014 G.f. = 1 + 2*q + 4*q^2 - 2*q^4 - 8*q^5 + 4*q^8 - 4*q^9 - 4*q^10 + 4*q^13 + ...
%t A256014 a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, q^3]^4 / (EllipticTheta[ 4, 0, q] EllipticTheta[ 4, 0, q^9]), {q, 0, n}];
%o A256014 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A) * eta(x^3 + A)^8 * eta(x^18 + A) / (eta(x + A)^2 * eta(x^6 + A)^4 * eta(x^9 + A)^2), n))};
%o A256014 (PARI) {a(n) = if( n<1, n==0, 2^(n%3) * (-1)^(n\3) * sumdiv(n, d, [0, 1, 2, -1][d%4 + 1] * if(d%9, 1, 4) * (-1)^((d%8==6) + n+d)))};
%Y A256014 Cf. A002175, A104794, A121444, A122856, A122865, A256280, A258277, A258278.
%K A256014 sign
%O A256014 0,2
%A A256014 _Michael Somos_, Jun 03 2015