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%I #15 Mar 12 2021 22:24:46
%S 1,1,-2,-1,0,-2,1,-1,-1,4,2,2,0,-1,2,1,-2,-3,-2,0,0,-4,-2,2,1,0,4,-1,
%T 0,2,-4,1,0,2,0,1,0,2,2,-4,2,0,2,-2,0,2,0,-2,1,-1,-4,0,0,4,-4,1,-2,-2,
%U 2,-2,0,0,-1,-1,2,-4,-2,2,0,0,0,3,2,-2,2,2,0,6
%N Expansion of q * psi(-q) * phi(q) * psi(-q^7) in powers of q 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="/A215556/b215556.txt">Table of n, a(n) for n = 1..5000</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^2)^4 * eta(q^7) * eta(q^28) / (eta(q) * eta(q^4) * eta(q^14)) in powers of q.
%F Euler transform of period 28 sequence [ 1, -3, 1, -2, 1, -3, 0, -2, 1, -3, 1, -2, 1, -3, 1, -2, 1, -3, 1, -2, 0, -3, 1, -2, 1, -3, 1, -3, ...].
%F G.f. is a period 1 Fourier series which satisfies f(-1 / (28 t)) = 392^(1/2) (t/i)^(3/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A159813.
%F a(7*n) = A159813(n).
%e G.f. = q + q^2 - 2*q^3 - q^4 - 2*q^6 + q^7 - q^8 - q^9 + 4*q^10 + 2*q^11 + 2*q^12 + ...
%t a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q] EllipticTheta[ 2, Pi/4, q^(1/2)] EllipticTheta[ 2, Pi/4, q^(7/2)] / 2, {q, 0, n}]; (* _Michael Somos_, Aug 27 2015 *)
%o (PARI) {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x^2 + A)^4 * eta(x^7 + A) * eta(x^28 + A) / (eta(x + A) * eta(x^4 + A) * eta(x^14 + A)), n))};
%Y Cf. A159813.
%K sign
%O 1,3
%A _Michael Somos_, Aug 15 2012
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