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A134416 Expansion of eta(q^4)^2 / (eta(q^2) * eta(q)^6) in powers of q. 3
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 (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

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

M. 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 A117999 A234617

Adjacent sequences:  A134413 A134414 A134415 * A134417 A134418 A134419

KEYWORD

nonn

AUTHOR

Michael Somos, Oct 26 2007

STATUS

approved

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Last modified June 20 09:12 EDT 2019. Contains 324234 sequences. (Running on oeis4.)