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

0,2

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

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

Example

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

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 *)

Prog

(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

Crossrefs

Cf. A134414, A245643.

Sequence in context: A172132 A011856 A276041 * A266976 A352739 A117999

Adjacent sequences: A134413 A134414 A134415 * A134417 A134418 A134419

Keyword

nonn

Author

Michael Somos, Oct 26 2007

Status

approved