OFFSET
0,2
COMMENTS
REFERENCES
D. Ford, J. McKay and S. P. Norton, More on replicable functions, Commun. Algebra 22, No. 13, 5175-5193 (1994).
LINKS
Seiichi Manyama, 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
Eric Weisstein's World of Mathematics, Infinite Product
FORMULA
Expansion of q^(1/3) * (eta(q^2)^2 / (eta(q) * eta(q^4)))^8 in powers of q.
Euler transform of period 4 sequence [8, -8, 8, 0, ...].
Given g.f. A(x), B(q) = A(q^3) / q satisfies 0 = f(B(q), B(q^2)) where f(u, v) = u*v*(u^3+v^3) -(u*v)^3 + 15*(u*v)^2 - 32*u*v + 16.
G.f.: (Product_{k>0} (1 + x^(2*k-1)))^8.
a(n) ~ exp(2*Pi*sqrt(n/3)) / (2 * 3^(1/4) * n^(3/4)). - Vaclav Kotesovec, Aug 27 2015
Expansion of chi(x)^8 in powers of x where chi() is a Ramanujan theta function. - Michael Somos, Sep 12 2017
G.f.: exp(8*Sum_{k>=1} x^k/(k*(1 - (-x)^k))). - Ilya Gutkovskiy, Jun 07 2018
EXAMPLE
T12D = 1/q + 8*q^2 + 28*q^5 + 64*q^8 + 134*q^11 + 288*q^14 + 568*q^17 + ...
MATHEMATICA
nmax = 50; CoefficientList[Series[Product[(1 + x^(2*k+1))^8, {k, 0, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 27 2015 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ -x, x^2]^8, {x, 0, n}]; (* Michael Somos, Sep 12 2017 *)
PROG
(PARI) {a(n) = my(A); if(n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A)^2 / (eta(x + A) * eta(x^4 + A)))^8, n))};
(PARI) {a(n) = my(A); if(n<0, 0, A = x * O(x^n); polcoeff( prod(k=1, (n+1)\2, 1 + x^(2*k-1), 1 + A)^8, n))};
CROSSREFS
KEYWORD
nonn
AUTHOR
Michael Somos, Dec 02 2004
STATUS
approved