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A096727 Expansion of eta(q)^8 / eta(q^2)^4 in powers of q.

%I A096727 #39 Feb 16 2025 08:32:53
%S A096727 1,-8,24,-32,24,-48,96,-64,24,-104,144,-96,96,-112,192,-192,24,-144,
%T A096727 312,-160,144,-256,288,-192,96,-248,336,-320,192,-240,576,-256,24,
%U A096727 -384,432,-384,312,-304,480,-448,144,-336,768,-352,288,-624,576,-384,96,-456,744,-576,336,-432,960,-576,192
%N A096727 Expansion of eta(q)^8 / eta(q^2)^4 in powers of q.
%C A096727 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A096727 Seiichi Manyama, <a href="/A096727/b096727.txt">Table of n, a(n) for n = 0..10000</a>
%H A096727 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A096727 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%H A096727 K. S. Williams, <a href="http://www.jstor.org/stable/10.4169/amer.math.monthly.120.04.329">The parents of Jacobi's four squares theorem are unique</a>, Amer. Math. Monthly, 120 (2013), 329-345.
%F A096727 a(n) =  -8*sigma(n) + 48*sigma(n/2) - 64*sigma(n/4) for n>0, where sigma(n) = A000203(n) if n is an integer, otherwise 0.
%F A096727 Euler transform of period 2 sequence [ -8, -4, ...].
%F A096727 G.f.: Prod_{k>0} (1 - x^k)^8 / (1 - x^(2k))^4 = 1 + Sum_{k>0} k * (-8 * x^k / (1 - x^k) + 48 * x^(2*k)  /(1 - x^(2*k)) - 64 * x^(4*k)/(1 - x^(4*k))).
%F A096727 G.f. theta_4(q)^4 = (Sum_{k} (-q)^(k^2))^4.
%F A096727 Expansion of phi(-q)^4 in powers of q where phi() is a Ramanujan theta function. - _Michael Somos_, Nov 01 2006
%F A096727 G.f. A(x) satisfies 0 = f(A(x), A(x^3), A(x^9)) where f(u, v, w) = v^4 - 30*u*v^2*w + 12*u*v*w * (u + 9*w) - u*w * (u^2 + 9*w*u + 81*w^2).
%F A096727 a(n) = (-1)^n * A000118(n). a(n) = 8 * A109506(n) unless n=0. a(2*n) = A004011(n). a(2*n + 1) = -A005879(n).
%F A096727 a(0) = 1, a(n) = -(8/n)*Sum_{k=1..n} A002131(k)*a(n-k) for n > 0. - _Seiichi Manyama_, May 02 2017
%e A096727 G.f. = 1 - 8*q + 24*q^2 - 32*q^3 + 24*q^4 - 48*q^5 + 96*q^6 - 64*q^7 + 24*q^8 - ...
%t A096727 CoefficientList[ Series[1 + Sum[k(-8x^k/(1 - x^k) + 48x^(2k)/(1 - x^(2k)) - 64x^(4k)/(1 - x^(4k))), {k, 1, 60}], {x, 0, 60}], x] (* _Robert G. Wilson v_, Jul 14 2004 *)
%t A096727 a[ n_] := With[{m = InverseEllipticNomeQ @ q}, SeriesCoefficient[ q Dt[ Log @ m, q], {q, 0, n}]]; (* _Michael Somos_, Sep 06 2012 *)
%t A096727 a[ n_] := (-1)^n SquaresR[ 4, n]; (* _Michael Somos_, Jun 12 2014 *)
%t A096727 a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, q]^4, {q, 0, n}]; (* _Michael Somos_, Jun 12 2014 *)
%t A096727 QP = QPochhammer; s = QP[q]^8/QP[q^2]^4 + O[q]^60; CoefficientList[s, q] (* _Jean-François Alcover_, Nov 23 2015 *)
%o A096727 (PARI) {a(n) = if( n<1, n==0, 8 * (-1)^n * sumdiv( n, d, if( d%4, d)))};
%o A096727 (PARI) {a(n) = local(A); if( n<0, 0, A = x *O (x^n); polcoeff( eta(x + A)^8 / eta(x^2 + A)^4, n))};
%o A096727 (Sage) A = ModularForms( Gamma0(4), 2, prec=57) . basis(); A[0] - 8*A[1]; # _Michael Somos_, Jun 12 2014
%o A096727 (Magma) A := Basis( ModularForms( Gamma0(4), 2), 57); A[1] - 8*A[2]; /* _Michael Somos_, Aug 21 2014 */
%o A096727 (Julia) # JacobiTheta4 is defined in A002448.
%o A096727 A096727List(len) = JacobiTheta4(len, 4)
%o A096727 A096727List(57) |> println # _Peter Luschny_, Mar 12 2018
%Y A096727 Cf. A000118, A002131, A004011, A005879, A109506.
%K A096727 sign
%O A096727 0,2
%A A096727 _Michael Somos_, Jul 06 2004