A134416 Expansion of eta(q^4)^2 / (eta(q^2) * eta(q)^6) in powers of q.

1, 6, 28, 104, 342, 1016, 2808, 7296, 18044, 42750, 97656, 215992, 464360, 973176, 1993328, 3998592, 7870038, 15221232, 28968084, 54311736, 100421688, 183281904, 330468216, 589084288, 1038850488, 1813500030, 3135518440, 5372110496, 9124793472, 15371832424

Offset: 0

Author

Michael Somos, Oct 26 2007

Keywords

• nonn

Comments

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

Examples

			G.f. = 1 + 6*q + 28*q^2 + 104*q^3 + 342*q^4 + 1016*q^5 + 2808*q^6 + 7296*q^7 + ...
		

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

Crossrefs

Cf. A134414, A245643.

Programs

Mathematica

nmax = 40; CoefficientList[Series[Product[(1 + x^k) * (1 + x^(2*k))^2 / (1 - x^k)^5, {k, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 07 2015 *)
a[ n_] := SeriesCoefficient[ 1 / (EllipticTheta[ 4, 0 , q]^3 EllipticTheta[ 4, 0, q^2]^2), {q, 0, n}]; (* Michael Somos, Oct 16 2015 *)
a[ n_] := SeriesCoefficient[ 1 / (EllipticTheta[ 3, 0, q] EllipticTheta[ 4, 0, q]^4), {q, 0, n}]; (* Michael Somos, Oct 16 2015 *)
a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q]^3 / EllipticTheta[ 4, 0, q^2]^8, {q, 0, n}]; (* Michael Somos, Oct 16 2015 *)
QP = QPochhammer; s = QP[q^4]^2/(QP[q^2]*QP[q]^6) + O[q]^30; CoefficientList[s, q] (* Jean-François Alcover, Nov 27 2015 *)

PARI

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^4 + A)^2 / (eta(x^2 + A) * eta(x + A)^6), n))};

PARI

q='q+O('q^99); Vec(eta(q^4)^2/(eta(q^2)*eta(q)^6)) \\ Altug Alkan, Apr 16 2018

Formula

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

G.f.: Product_{k>0} (1 + x^k) * (1 + x^(2*k))^2 / (1 - x^k)^5. [corrected by Vaclav Kotesovec, Sep 07 2015]

a(n) ~ exp(2*Pi*sqrt(n))/(32*n^2). - Vaclav Kotesovec, Sep 07 2015

-2 * a(n) = A134414(4*n).

Expansion of psi(q^2) / f(-q)^6 = phi(q)^3 / phi(-q^2)^8 = 1 / (phi(-q)^3 * phi(-q^2)^2) = 1 / (phi(q) * phi(-q)^4) in powers of q where phi(), psi(), f() are Ramanujan theta functions. - Michael Somos, Oct 16 2015

G.f. is a period 1 Fourier series which satisfies f(-1 / (4 t)) = 2^(-13/2) (t/i)^(-5/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A134414. - Michael Somos, Oct 16 2015

Convolution inverse is A245643. - Michael Somos, Oct 16 2015