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A096727 Expansion of eta(q)^8 / eta(q^2)^4 in powers of q. 11
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

M. 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-Fran├žois 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 */

CROSSREFS

Cf. A000118, A002131, A004011, A005879, A109506.

Sequence in context: A162829 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 July 22 10:33 EDT 2017. Contains 289669 sequences.