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%I A193514 #14 Feb 16 2025 08:33:15
%S A193514 1,-4,4,2,-4,0,4,-8,4,2,0,0,2,-8,8,0,-4,0,4,-8,0,4,0,0,4,-4,8,2,-8,0,
%T A193514 0,-8,4,0,0,0,2,-8,8,4,0,0,8,-8,0,0,0,0,2,-12,4,0,-8,0,4,0,8,4,0,0,0,
%U A193514 -8,8,4,-4,0,0,-8,0,0,0,0,4,-8,8,2,-8,0,8,-8,0,2,0,0,4,0,8,0,0,0,0,-16,0,4,0,0,4,-8
%N A193514 Expansion of phi(-q)^2 * phi(-q^9) / phi(-q^3) in powers of q where phi() is a Ramanujan theta function.
%C A193514 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A193514 G. C. Greubel, <a href="/A193514/b193514.txt">Table of n, a(n) for n = 0..1000</a>
%H A193514 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A193514 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A193514 Expansion of (-2 * a(q) +2 * a(q^2) +3 * a(q^3)) / 3 = b(q) * (b(q) + 2 * b(q^2)) / (3 * b(q^2)) in powers of q where a(), b() are cubic AGM functions.
%F A193514 Expansion of eta(q)^4 * eta(q^6) * eta(q^9)^2 / (eta(q^2)^2 * eta(q^3)^2 * eta(q^18)) in powers of q.
%F A193514 Euler transform of period 18 sequence [ -4, -2, -2, -2, -4, -1, -4, -2, -4, -2, -4, -1, -4, -2, -2, -2, -4, -2, ...].
%F A193514 Moebius transform is period 18 sequence [ -4, 8, 6, -8, 4, -6, -4, 8, 0, -8, 4, 6, -4, 8, -6, -8, 4, 0, ...].
%F A193514 G.f. is a period 1 Fourier series which satisfies f(-1 / (18 t)) = 432^(1/2) (t / i) g(t) where q = exp(2 Pi i t) and g() is g.f. for A193426.
%F A193514 a(3*n) = A123330(n). a(3*n + 1) = -4 * A033687(n). a(6*n + 1) = -4 * A097195(n). a(6*n + 2) = 4 * A033687(n). a(6*n + 3) = 2 * A033762(n). a(6*n + 4) = 4 * A033687(n). a(8*n + 2) = 4 * A112604(n). a(8*n + 6) = 4 * A112605(n). a(6*n + 5) = 0. a(4*n) = a(n).
%e A193514 G.f. = 1 - 4*q + 4*q^2 + 2*q^3 - 4*q^4 + 4*q^6 - 8*q^7 + 4*q^8 + 2*q^9 + 2*q^12 + ...
%t A193514 a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, q]^2 EllipticTheta[ 4, 0, q^9] / EllipticTheta[ 4, 0, q^3], {q, 0, n}];
%o A193514 (PARI) {a(n) = if( n<1, n==0, 2 * if( n%3==1, -2, 1) * sumdiv( n, d, -(-1)^d * kronecker( -3, d)))};
%o A193514 (PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^4 * eta(x^6 + A) * eta(x^9 + A)^2 / (eta(x^2 + A)^2 * eta(x^3 + A)^2 * eta(x^18 + A)), n))};
%Y A193514 Cf. A033687, A033762, A097195, A112604, A112605.
%K A193514 sign,changed
%O A193514 0,2
%A A193514 _Michael Somos_, Jul 29 2011

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