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 A210065 Expansion of phi(q^2) / phi(q) in powers of q where phi() is a Ramanujan theta function. 2
 1, -2, 6, -12, 22, -40, 68, -112, 182, -286, 440, -668, 996, -1464, 2128, -3056, 4342, -6116, 8538, -11820, 16248, -22176, 30068, -40528, 54308, -72378, 95976, -126648, 166352, -217560, 283344, -367552, 474998, -611624, 784812, -1003712, 1279562, -1626216 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700). LINKS G. C. Greubel, Table of n, a(n) for n = 0..2500 Michael Somos, Introduction to Ramanujan theta functions Eric Weisstein's World of Mathematics, Ramanujan Theta Functions FORMULA Expansion of (eta(q) / eta(q^8))^2 * (eta(q^4) / eta(q^2))^7 in powers of q. Euler transform of period 8 sequence [-2, 5, -2, -2, -2, 5, -2, 0, ...]. G.f. is a period 1 Fourier series which satisfies f(-1 / (8 t)) = 2^(-1/2) * g(t) where q = exp(2 Pi i t) and g() is g.f. for A080015. a(n) = (-1)^n * A208850(n). Convolution inverse of A080015. a(n) ~ (-1)^n * exp(sqrt(n)*Pi) / (8*n^(3/4)). - Vaclav Kotesovec, Nov 17 2017 EXAMPLE G.f. = 1 - 2*q + 6*q^2 - 12*q^3 + 22*q^4 - 40*q^5 + 68*q^6 - 112*q^7 + 182*q^8 + ... MATHEMATICA nmax = 40; CoefficientList[Series[Product[((1 - x^k) / (1 - x^(8*k)))^2 * (1 + x^(2*k))^7, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 17 2017 *) eta[q_]:= q^(1/24)*QPochhammer[q]; CoefficientList[Series[(eta[q]/ eta[q^8])^2*(eta[q^4]/eta[q^2])^7, {q, 0, 50}], q] (* G. C. Greubel, Aug 11 2018 *) PROG (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x + A) / eta(x^8 + A))^2 * (eta(x^4 + A) / eta(x^2 + A))^7, n))}; CROSSREFS Cf. A080015, A208850. Sequence in context: A168193 A182977 A116658 * A208850 A131520 A086953 Adjacent sequences:  A210062 A210063 A210064 * A210066 A210067 A210068 KEYWORD sign AUTHOR Michael Somos, Mar 16 2012 STATUS approved

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Last modified July 24 01:41 EDT 2021. Contains 346269 sequences. (Running on oeis4.)