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%I #13 Sep 08 2022 08:46:11
%S 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 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 -4,0,0,-2,16,0,0,8,0,0,0,4,-4,8,0,0,0,0,0,8
%N Expansion of phi(q^3)^4 / (phi(q) * phi(q^9)) in powers of q where phi() is a Ramanujan theta function.
%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H G. C. Greubel, <a href="/A256280/b256280.txt">Table of n, a(n) for n = 0..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 (eta(q) * eta(q^4) * eta(q^6)^10 * eta(q^9) * eta(q^36))^2 / (eta(q^2)^5 * eta(q^3)^8 * eta(q^12)^8 * eta(q^18)^5) in powers of q.
%F Euler transform of period 36 sequence [ -2, 3, 6, 1, -2, -9, -2, 1, 4, 3, -2, -3, -2, 3, 6, 1, -2, -6, -2, 1, 6, 3, -2, -3, -2, 3, 4, 1, -2, -9, -2, 1, 6, 3, -2, -2, ...].
%F G.f. is a period 1 Fourier series which satisfies f(-1 / (36 t)) = 6 (t/i) f(t) where q = exp(2 Pi i t).
%F a(3*n + 1) = -2 * A122865(n). a(3*n + 2) = 4 * A122856(n). a(4*n + 3) = 0. a(4*n) = a(n). a(9*n) = A004018(n). a(9*n + 3) = a(9*n + 6) = 0. a(12*n + 1) = -2 * A002175(n). a(12*n + 5) = 8 * A121444(n).
%e 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 a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q^3]^4 / (EllipticTheta[ 3, 0, q] EllipticTheta[ 3, 0, q^9]), {q, 0, n}];
%o (PARI) {a(n) = if( n<1, n==0, (-2)^(n%3) * sumdiv(n, d, [0, 1, 2, -1][d%4 + 1] * if(d%9, 1, 4) * (-1)^((d%8==6) + n+d)))};
%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x + A) * eta(x^4 + A) * eta(x^6 + A)^10 * eta(x^9 + A) * eta(x^36 + A))^2 / (eta(x^2 + A)^5 * eta(x^3 + A)^8 * eta(x^12 + A)^8 * eta(x^18 + A)^5), n))};
%o (Magma) A := Basis( ModularForms( Gamma1(36), 1), 91); A[1] - 2*A[2] + 4*A[3] - 2*A[5] + 8*A[6] + 4*A[9] + 4*A[10] - 4*A[11] - 4*A[14] - 2*A[17] + 8*A[18] + 4*A[19];
%Y Cf. A002175, A004018, A121444, A122856, A122865.
%K sign
%O 0,2
%A _Michael Somos_, Jun 02 2015
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