A123861 Expansion of (f(q) * f(q^3) / (f(-q) * f(-q^3)))^2 in powers of q where f() is a Ramanujan theta function.

1, 4, 8, 20, 48, 88, 168, 320, 544, 932, 1584, 2544, 4080, 6488, 9984, 15288, 23232, 34568, 51144, 75152, 108832, 156736, 224352, 317728, 447648, 627292, 871856, 1206068, 1660416, 2271032, 3092976, 4194464, 5657728, 7602096, 10175760

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

Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], 2015-2016.

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

Expansion of phi(q) * phi(q^3) / (phi(-q) * phi(-q^3)) in powers of q where phi() is a Ramanujan theta function. - Michael Somos, Aug 31 2014

Expansion of eta(q^2)^6 * eta(q^6)^6 / (eta(q)^4 * eta(q^3)^4 * eta(q^4)^2 * eta(q^12)^2) in powers of q. - Michael Somos, Aug 31 2014

Euler transform of period 12 sequence [4, -2, 8, 0, 4, -4, 4, 0, 8, -2, 4, 0, ...].

G.f. A(q) satisfies 0 = f(A(q), A(q^2)) where f(u, v) = (u - 1)^2 - 4 * u*v * (v - 1).

Let g.f. A(x) = u, then B(x) = u * (u-1) / 4, B(x^2) = ((u-1) / 4)^2 / u where B(x) is the g.f. for A123653.

G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = (1/4) * g(t) where q = exp(2 Pi i t) and g() is the g.f. for A187197. - Michael Somos, Aug 31 2014

a(n) = 4 * A123647(n) unless n=0.

a(n) ~ exp(2*Pi*sqrt(n/3)) / (8 * 3^(1/4) * n^(3/4)). - Vaclav Kotesovec, Oct 13 2015

Example

G.f. = 1 + 4*q + 8*q^2 + 20*q^3 + 48*q^4 + 88*q^5 + 168*q^6 + 320*q^7 + ...

Mathematica

a[ n_] := SeriesCoefficient[ (QPochhammer[ -q] QPochhammer[ -q^3] / (QPochhammer[ q] QPochhammer[ q^3]))^2, {q, 0, n}]; (* Michael Somos, Aug 31 2014 *)
nmax=60; CoefficientList[Series[Product[(1+x^k)^6 * (1-x^k)^2 * (1+x^(3*k))^4 / ((1-x^(4*k))^2 * (1+x^(6*k))^2), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Oct 13 2015 *)

Prog

(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^6 * eta(x^6 + A)^6 / (eta(x + A)^4 * eta(x^3 + A)^4 * eta(x^4 + A)^2 * eta(x^12 + A)^2), n))};

Crossrefs

Cf. A123647, A187197.

Sequence in context: A190589 A009916 A203167 * A357287 A371596 A115099

Adjacent sequences: A123858 A123859 A123860 * A123862 A123863 A123864

Keyword

nonn

Author

Michael Somos, Oct 14 2006

Status

approved