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A034950 Expansion of eta(8z)*eta(16z)*theta_3(2z).

%I A034950 #42 Feb 16 2025 08:32:37
%S A034950 1,2,0,0,1,-2,0,0,-4,-2,0,0,-3,0,0,0,4,-4,0,0,0,6,0,0,1,4,0,0,4,2,0,0,
%T A034950 0,-2,0,0,4,-2,0,0,-3,2,0,0,-4,-4,0,0,-4,2,0,0,-8,-6,0,0,8,-4,0,0,1,
%U A034950 -4,0,0,-4,6,0,0,0,2,0,0,0,-2,0,0,4,8,0,0,0,6,0,0,5,-2,0,0,4,-2,0,0,8,4,0,0,-4,-8,0,0,-4,8,0,0,4
%N A034950 Expansion of eta(8z)*eta(16z)*theta_3(2z).
%C A034950 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A034950 Seiichi Manyama, <a href="/A034950/b034950.txt">Table of n, a(n) for n = 0..10000</a> (terms 0..1000 from G. C. Greubel)
%H A034950 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 A034950 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A034950 J. B. Tunnell, <a href="http://dx.doi.org/10.1007/BF01389327">A classical Diophantine problem and modular forms of weight 3/2</a>, Invent. Math., 72 (1983), 323-334.
%H A034950 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A034950 Euler transform of period 8 sequence [2, -3, 2, -2, 2, -3, 2, -3, ...]. - _Michael Somos_, Feb 16 2006
%F A034950 Expansion of q^(-1/2) * eta(q^2)^5 * eta(q^8) / (eta(q)^2 * eta(q^4)) in powers of q. - _Michael Somos_, Feb 16 2006
%F A034950 Expansion of psi(x)^2 * psi(-x^2) = phi(x) * psi(x^2) * psi(-x^2) = phi(x) * psi(x^4) * phi(-x^4) in powers of x where phi(), psi() are Ramanujan theta functions. - _Michael Somos_, Feb 18 2015
%F A034950 G.f.: Product_{k>0} (1 + x^k)^2 * (1 - x^(2*k))^3 * (1 + x^(4*k)). - _Michael Somos_, Feb 16 2006
%F A034950 2 * a(n) = A080963(2*n + 1). a(4*n + 2) = a(4*n + 3) = 0. - _Michael Somos_, Feb 18 2015
%F A034950 a(n) = A072069(n+1) - A072068(n+1)/2. - _Seichi Manymama_, Sep 30 2018
%e A034950 G.f. = 1 + 2*x + x^4 - 2*x^5 - 4*x^8 - 2*x^9 - 3*x^12 + 4*x^16 - 4*x^17 + ...
%e A034950 G.f. = q + 2*q^3 + q^9 - 2*q^11 - 4*q^17 - 2*q^19 - 3*q^25 + 4*q^33 - ...
%t A034950 a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, x] EllipticTheta[ 2, 0, x] EllipticTheta[ 2, Pi/4, x] / Sqrt[8 x], {x, 0, n}]; (* _Michael Somos_, Feb 18 2015 *)
%t A034950 QP = QPochhammer; s = QP[q^2]^5*(QP[q^8]/(QP[q]^2*QP[q^4])) + O[q]^105; CoefficientList[s, q] (* _Jean-François Alcover_, Nov 27 2015 *)
%o A034950 (PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^5 * eta(x^8 + A) / (eta(x + A)^2 * eta(x^4 + A)), n))}; /* _Michael Somos_, Feb 16 2006 */
%Y A034950 Cf. A072068, A072069, A080963.
%Y A034950 A bisection of A248394.
%K A034950 sign
%O A034950 0,2
%A A034950 _N. J. A. Sloane_