A259491 Expansion of (eta(q)^2 * eta(q^2) * eta(q^4)^3 / eta(q^8)^2)^2 in powers of q.

1, -4, 0, 16, -16, 8, 0, -96, 112, 44, 0, 176, -448, -88, 0, -32, 1136, -200, 0, -176, -2016, 384, 0, 224, 3136, 484, 0, -608, -5504, -792, 0, 640, 9328, -704, 0, 192, -12112, 648, 0, 352, 14112, 792, 0, -208, -21312, -88, 0, -2112, 31808, -932, 0, 800

Offset

0,2

Comments

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

Expansion of (phi(q) * phi(q^2) * phi(-q)^2)^2 in powers of q where phi() is a Ramanujan theta function.

Euler transform of period 8 sequence [ -4, -6, -4, -12, -4, -6, -4, -8, ...].

G.f.: Product_{k>0} ((1 - x^k)^4 * (1 + x^k)^2 * (1 + x^(2*k)) / (1 + x^(4*k))^2)^2.

a(2*n + 1) = -4 * A030211(n). a(4*n) = A035016(n). a(4*n + 2) = 0.

Convolution square of A131999.

Example

G.f. = 1 - 4*q + 16*q^3 - 16*q^4 + 8*q^5 - 96*q^7 + 112*q^8 + 44*q^9 + ...

Mathematica

a[ n_] := SeriesCoefficient[ (QPochhammer[ q]^2 QPochhammer[ q^2] QPochhammer[ q^4]^3 / QPochhammer[ q^8]^2)^2, {q, 0, n}];
a[ n_] := SeriesCoefficient[ (EllipticTheta[ 3, 0, q] EllipticTheta[ 3, 0, q^2] EllipticTheta[ 4, 0, q]^2)^2, {q, 0, n}];
a[ n_] := SeriesCoefficient[ (EllipticTheta[ 4, 0, q] EllipticTheta[ 4, 0, q^2] EllipticTheta[ 4, 0, q^4]^2)^2, {q, 0, n}];

Prog

(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x + A)^2 * eta(x^2 + A) * eta(x^4 + A)^3 / eta(x^8 + A)^2)^2, n))};
(Magma) A := Basis( ModularForms( Gamma1(8), 4), 52); A[1] - 4*A[2] + 16*A[4] - 16*A[5] + 8*A[6] - 96*A[8] + 112*A[9];

Crossrefs

Cf. A030211, A035016, A131999.

Sequence in context: A230280 A030212 A167359 * A292180 A261979 A007216

Adjacent sequences: A259488 A259489 A259490 * A259492 A259493 A259494

Keyword

sign

Author

Michael Somos, Jun 28 2015

Status

approved