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
Keywords
Examples
G.f. = 1 + 4*q + 8*q^2 + 20*q^3 + 48*q^4 + 88*q^5 + 168*q^6 + 320*q^7 + ...
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
Programs
-
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 *) -
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))};
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
Comments