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 A096727 Expansion of eta(q)^8 / eta(q^2)^4 in powers of q. 17
 1, -8, 24, -32, 24, -48, 96, -64, 24, -104, 144, -96, 96, -112, 192, -192, 24, -144, 312, -160, 144, -256, 288, -192, 96, -248, 336, -320, 192, -240, 576, -256, 24, -384, 432, -384, 312, -304, 480, -448, 144, -336, 768, -352, 288, -624, 576, -384, 96, -456, 744, -576, 336, -432, 960, -576, 192 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700). LINKS Seiichi Manyama, Table of n, a(n) for n = 0..10000 Michael Somos, Introduction to Ramanujan theta functions Eric Weisstein's World of Mathematics, Ramanujan Theta Functions K. S. Williams, The parents of Jacobi's four squares theorem are unique, Amer. Math. Monthly, 120 (2013), 329-345. FORMULA 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. Euler transform of period 2 sequence [ -8, -4, ...]. 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))). G.f. theta_4(q)^4 = (Sum_{k} (-q)^(k^2))^4. Expansion of phi(-q)^4 in powers of q where phi() is a Ramanujan theta function. - Michael Somos, Nov 01 2006 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). 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). a(0) = 1, a(n) = -(8/n)*Sum_{k=1..n} A002131(k)*a(n-k) for n > 0. - Seiichi Manyama, May 02 2017 EXAMPLE 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 - ... MATHEMATICA 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 *) a[ n_] := With[{m = InverseEllipticNomeQ @ q}, SeriesCoefficient[ q Dt[ Log @ m, q], {q, 0, n}]]; (* Michael Somos, Sep 06 2012 *) a[ n_] := (-1)^n SquaresR[ 4, n]; (* Michael Somos, Jun 12 2014 *) a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, q]^4, {q, 0, n}]; (* Michael Somos, Jun 12 2014 *) QP = QPochhammer; s = QP[q]^8/QP[q^2]^4 + O[q]^60; CoefficientList[s, q] (* Jean-François Alcover, Nov 23 2015 *) PROG (PARI) {a(n) = if( n<1, n==0, 8 * (-1)^n * sumdiv( n, d, if( d%4, d)))}; (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))}; (Sage) A = ModularForms( Gamma0(4), 2, prec=57) . basis(); A[0] - 8*A[1]; # Michael Somos, Jun 12 2014 (MAGMA) A := Basis( ModularForms( Gamma0(4), 2), 57); A[1] - 8*A[2]; /* Michael Somos, Aug 21 2014 */ (Julia) # JacobiTheta4 is defined in A002448. A096727List(len) = JacobiTheta4(len, 4) A096727List(57) |> println # Peter Luschny, Mar 12 2018 CROSSREFS Cf. A000118, A002131, A004011, A005879, A109506. Sequence in context: A303796 A175368 A000118 * A028660 A028644 A227175 Adjacent sequences:  A096724 A096725 A096726 * A096728 A096729 A096730 KEYWORD sign AUTHOR Michael Somos, Jul 06 2004 STATUS approved

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Last modified April 19 13:00 EDT 2021. Contains 343114 sequences. (Running on oeis4.)