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%I #27 Mar 12 2021 22:24:42
%S 1,2,2,4,1,-2,2,-4,-2,2,-8,-4,-1,-4,-6,0,-4,-8,10,-4,-6,6,2,8,9,-4,-6,
%T 4,4,14,2,4,4,10,8,-12,14,-2,8,8,-11,-6,-4,12,-2,-8,0,-4,-2,-2,-6,4,
%U -16,-2,-6,-20,2,8,2,-8,-7,-12,-12,-16,12,-6,-8,8,10,-10,-16,4,-12,18,18,-4,-2,0,18,12,-16,2,-8,20,-9,2,18,-4,28,-6,2
%N Expansion of eta(8z)*eta(16z)*theta_3(2z)*theta_3(4z).
%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H G. C. Greubel, <a href="/A034951/b034951.txt">Table of n, a(n) for n = 0..1000</a>
%H Johann Cigler, <a href="https://homepage.univie.ac.at/johann.cigler/preprints/losanitsch3.pdf">Some Pascal-like triangles</a>, 2018.
%H Ken Ono and Christopher Skinner, <a href="http://www.jstor.org/stable/121015">Fourier Coefficients of Half-Integral Weight Modular Forms Modulo l</a>, Ann. Math., 147 (1998), 453-470.
%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 G.f.: Product_{k>0} (1 - x^(2*k)) * (1 + x^k)^2 * (1 - x^(4*k))^3 / (1 + x^(4*k)). - _Michael Somos_, Sep 21 2005
%F Euler transform of period 8 sequence [2, -1, 2, -5, 2, -1, 2, -4, ...]. - _Michael Somos_, Sep 21 2005
%F Expansion of q^(-1/2) * (eta(q^2)^3 * eta(q^4)^4) / (eta(q)^2 * eta(q^8)) in powers of q. - _Michael Somos_, Sep 21 2005
%F Expansion of phi(x) * f(x^2)^2 * f(-x^8) = psi(x)^2 * f(x^2) * f(-x^4) = psi(x)^2 * psi(-x^2) * phi(x^2) = psi(x^2)^2 * phi(x) * phi(-x^4) = psi(x)^2 * psi(x^2) * phi(-x^4) in powers of x where phi(), psi(), f() are Ramanujan theta functions. - _Michael Somos_, Jul 07 2014
%F a(31*n + 15) = 0 unless n == 15 (mod 31). a(961*n + 480) = -31 * a(n). - _Michael Somos_, Jul 07 2014
%e G.f. = 1 + 2*x + 2*x^2 + 4*x^3 + x^4 - 2*x^5 + 2*x^6 - 4*x^7 - 2*x^8 + ...
%e G.f. = q + 2*q^3 + 2*q^5 + 4*q^7 + q^9 - 2*q^11 + 2*q^13 - 4*q^15 - 2*q^17 + ...
%t a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, 0, x]^2 EllipticTheta[ 3, 0, x] EllipticTheta[ 4, 0, x^4] / (4 x^(1/2)), {x, 0, n}];
%t QP = QPochhammer; s = (QP[q^2]^3*QP[q^4]^4)/(QP[q]^2*QP[q^8]) + O[q]^90; CoefficientList[s, q] (* _Jean-François Alcover_, Nov 14 2015, adapted from PARI *)
%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^3 * eta(x^4 + A)^4 / (eta(x + A)^2 * eta(x^8 + A)), n))}; /* _Michael Somos_, Sep 21 2005 */
%K sign
%O 0,2
%A _N. J. A. Sloane_
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