A096727 Expansion of eta(q)^8 / eta(q^2)^4 in powers of q.
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
Offset: 0
Keywords
Examples
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 - ...
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.
Programs
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Julia
# JacobiTheta4 is defined in A002448. A096727List(len) = JacobiTheta4(len, 4) A096727List(57) |> println # Peter Luschny, Mar 12 2018
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Magma
A := Basis( ModularForms( Gamma0(4), 2), 57); A[1] - 8*A[2]; /* Michael Somos, Aug 21 2014 */
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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 *) -
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
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
Comments