OFFSET
0,2
COMMENTS
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1000
D. Ford, J. McKay and S. P. Norton, More on replicable functions, Commun. Algebra 22, No. 13, 5175-5193 (1994).
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
FORMULA
Expansion of (chi(-x^3) / chi(-x))^4 in powers of x where chi() is a Ramanujan theta function.
Expansion of q^(1/3) * c(q) * b(q^2) / (b(q) * c(q^2)) in powers of q where b(), c() are cubic AGM theta functions.
Expansion of q^(1/3) * (eta(q^2) * eta(q^3) / (eta(q) * eta(q^6)))^4 in powers of q.
Given g.f. A(x), then B(x) = A(x^3) / x satisfies 0 = f(B(x), B(x^2)) where f(u, v) = 8 * (u * v)^2 - (1 + u * v) * (u^2 - v) * (v^2 - u).
Given g.f. A(x), then B(x) = A(x^3) / x satisfies 0 = f(B(x), B(x^2)) where f(u, v) = 9 * (u * v)^2 - (u - v^2 + u^2*v) * (v - u^2 + u*v^2).
Given g.f. A(x), then B(x) = A(x^3) / x satisfies 0 = f(B(x), B(x^2), B(x^4)) where f(u, v, w) = 8 * u * v * w - (u^2 - v) * (w^2 - v).
Given g.f. A(x), then B(x) = A(x^3) / x satisfies 0 = f(B(x), B(x^5)) where f(u, v) = u*v * (1 + 25 * u*v + u^2*v^2)^2 - (u^3 + v^3 + 10 * u*v * (1 + u*v))^2.
G.f. is a period 1 Fourier series which satisfies f(-1 / (54 t)) = f(t) where q = exp(2 Pi i t).
a(n) ~ exp(2*Pi*sqrt(2*n)/3) / (2^(3/4) * sqrt(3) * n^(3/4)). - Vaclav Kotesovec, Sep 10 2015
EXAMPLE
1 + 4*x + 10*x^2 + 20*x^3 + 35*x^4 + 60*x^5 + 100*x^6 + 164*x^7 + ...
T18d = 1/q + 4*q^2 + 10*q^5 + 20*q^8 + 35*q^11 + 60*q^14 + 100*q^17 + ...
MATHEMATICA
nmax = 50; CoefficientList[Series[Product[((1+x^(3*k-1))*(1+x^(3*k-2)))^4, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 10 2015 *)
QP = QPochhammer; s = (QP[q^2]*(QP[q^3]/(QP[q]*QP[q^6])))^4 + O[q]^40; CoefficientList[s, q] (* Jean-François Alcover, Nov 30 2015, adapted from PARI *)
PROG
(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A) * eta(x^3 + A) / (eta(x + A) * eta(x^6 + A)))^4, n))} /* Michael Somos, Mar 04 2012 */
CROSSREFS
KEYWORD
nonn
AUTHOR
N. J. A. Sloane, Nov 27 2000
STATUS
approved