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%I #12 Mar 12 2021 22:24:46
%S 1,2,3,6,4,4,7,2,8,10,4,10,9,6,8,10,4,8,16,8,9,12,8,12,20,6,8,10,8,18,
%T 11,12,8,20,12,8,20,6,20,26,8,8,15,10,16,18,12,16,20,10,16,16,8,24,24,
%U 8,21,26,8,20,20,14,8,28,16,10,28,10,24,22,8,16,17
%N Expansion of psi(x)^2 * phi(x^2) in powers of x where phi(), psi() are Ramanujan theta functions.
%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H G. C. Greubel, <a href="/A213625/b213625.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 q^(-1/4) * eta(q^2)^2 * eta(q^4)^5 / (eta(q)^2 * eta(q^8)^2) in powers of q.
%F Euler transform of period 8 sequence [ 2, 0, 2, -5, 2, 0, 2, -3, ...].
%F G.f. is a period 1 Fourier series which satisfies f(-1 / (32 t)) = 2^(3/2) (t/i)^(3/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A116597.
%F a(2*n) = A213622(n). a(2*n + 1) = 2 * A132969(n).
%e G.f. = 1 + 2*x + 3*x^2 + 6*x^3 + 4*x^4 + 4*x^5 + 7*x^6 + 2*x^7 + 8*x^8 + 10*x^9 + ...
%e G.f. = q + 2*q^5 + 3*q^9 + 6*q^13 + 4*q^17 + 4*q^21 + 7*q^25 + 2*q^29 + 8*q^33 + ...
%t a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, 0, x^(1/2)]^2 EllipticTheta[ 3, 0, x^2] / (4 x^(1/4)), {x, 0, n}];
%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^2 * eta(x^4 + A)^5 / (eta(x + A)^2 * eta(x^8 + A)^2), n))};
%Y Cf. A116597, A132969, A213622.
%K nonn
%O 0,2
%A _Michael Somos_, Jun 16 2012
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