OFFSET
1,2
COMMENTS
REFERENCES
R. Fricke, Die elliptischen Funktionen und ihre Anwendungen, Teubner, 1922, Vol. 2, see p. 375. Eqs. (20), (21), (24)
LINKS
Seiichi Manyama, Table of n, a(n) for n = 1..1000
Kevin Acres and David Broadhurst, Eta quotients and Rademacher sums, arXiv:1810.07478 [math.NT], 2018. See Table 1 p. 10.
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
FORMULA
Expansion of (eta(q^2) / eta(q^4))^2 * (eta(q^8) / eta(q))^4 in powers of q.
Expansion of Fricke tau_8(omega) / 16 in powers of q = exp(2 Pi i omega).
Expansion of elliptic (1/8) * (-1 + 1 / sqrt(1 - lambda(z)) = (1/8) * (-1 + 1 / k') in powers of the nome q = exp(Pi i z).
Expansion of ((phi(q) / phi(-q))^2 - 1) / 8 in powers of q where phi() is a Ramanujan theta function.
Elliptic j(z) = 256 * (x^4 + 8*x^3 + 20*x^2 + 16*x + 1)^3 / (x * (x + 4) * (x + 2)^2) where x = tau_8(z).
Euler transform of period 8 sequence [ 4, 2, 4, 4, 4, 2, 4, 0, ...].
G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = v - u^2 + 4 * v^2 + 8 * u * v + 32 * u * v^2.
G.f: x * Product_{k>0} (1 + x^k)^4 * (1 + x^(2*k))^2 * (1 + x^(4*k))^4.
a(n) ~ exp(sqrt(2*n)*Pi) / (64 * 2^(1/4) * n^(3/4)). - Vaclav Kotesovec, Sep 10 2015
EXAMPLE
G.f. = q + 4*q^2 + 12*q^3 + 32*q^4 + 78*q^5 + 176*q^6 + 376*q^7 + 768*q^8 + ...
MATHEMATICA
a[ n_] := SeriesCoefficient[ (1/4) (EllipticTheta[ 2, 0, q^2] / EllipticTheta[ 4, 0, q])^2, {q, 0, n}]; (* Michael Somos, Jun 13 2012 *)
a[ n_] := With[ {m = ModularLambda[ Log[q] / (Pi I)]}, SeriesCoefficient[ (1/8) (-1 + 1 / Sqrt[1 - m]), {q, 0, n}]]; (* Michael Somos, Jun 13 2012 *)
nmax = 50; CoefficientList[Series[Product[(1 + x^k)^4 * (1 + x^(2*k))^2 * (1 + x^(4*k))^4, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 10 2015 *)
QP = QPochhammer; s = (QP[q^2]/QP[q^4])^2*(QP[q^8]/QP[q])^4 + O[q]^40; CoefficientList[s, q] (* Jean-François Alcover, Nov 30 2015, adapted from PARI *)
PROG
(PARI) {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( (eta(x^2 + A) / eta(x^4 + A))^2 * (eta(x^8 + A) / eta(x + A))^4, n))};
CROSSREFS
KEYWORD
nonn
AUTHOR
Michael Somos, May 09 2005
STATUS
approved