A245643 Expansion of eta(q)^6 * eta(q^2) / eta(q^4)^2 in powers of q.

1, -6, 8, 16, -38, -16, 48, 64, -56, -150, 112, 112, -112, -80, 160, 192, -294, -288, 248, 304, -272, -160, 368, 320, -336, -726, 400, 448, -448, -240, 544, 640, -568, -864, 736, 608, -950, -400, 656, 832, -784, -1152, 864, 1008, -784, -496, 1184, 896, -1136

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)^4 = phi(-q)^3 * phi(-q^2)^2 = phi(-q^2)^8 / phi(q)^3 = f(-q)^6 / psi(q^2) in powers of q where phi(), psi(), f() are Ramanujan theta functions.

Euler transform of period 4 sequence [ -6, -7, -6, -5, ...].

G.f.: Product_{k>0} (1 - x^k)^6 * (1 - x^(2*k)) / (1 - x^(4*k))^2.

Convolution inverse of A134416.

G.f. is a period 1 Fourier series which satisfies f(-1 / (4 t)) = 8192^(1/2) (t/I)^(5/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A266575. - Michael Somos, Jan 03 2016

a(3*n + 2) = 8 * A263398(n). - Michael Somos, Oct 16 2015

Example

G.f. = 1 - 6*q + 8*q^2 + 16*q^3 - 38*q^4 - 16*q^5 + 48*q^6 + 64*q^7 + ...

Mathematica

a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q] EllipticTheta[ 4, 0, q]^4, {q, 0, n}];
a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, q]^3 EllipticTheta[ 4, 0, q^2]^2, {q, 0, n}];
a[ n_] := SeriesCoefficient[ 2 q^(1/4) QPochhammer[ q]^6 / EllipticTheta[ 2, 0, q], {q, 0, n}];

Prog

(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^6 * eta(x^2 + A) / eta(x^4 + A)^2, n))};
(Magma) A := Basis( ModularForms( Gamma1(4), 5/2), 49); A[1] - 6*A[2];

Crossrefs

Cf. A134416, A263398, A266575.

Sequence in context: A011989 A270821 A139452 * A299030 A127400 A315941

Adjacent sequences: A245640 A245641 A245642 * A245644 A245645 A245646

Keyword

sign

Author

Michael Somos, Sep 01 2014

Status

approved