OFFSET
0,3
COMMENTS
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 0..10000
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 (psi(q^2) * phi(q^17) - q^4 * phi(q) * psi(q^34)) / (f(-q) * f(-q^17)) in powers of q where phi(), psi(), f() are Ramanujan theta functions. - Michael Somos, Dec 11 2008
Expansion of (F(q) - q^4 / F(q)) / (chi(-q) * chi(-q^17))^2 in powers of q where F(q) = G(q^17) / G(q), G(q) = chi(q) * chi(-q^2) and chi() is a Ramanujan theta function. - Michael Somos, Dec 11 2008
G.f. is a period 1 Fourier series which satisfies f(-1 / (68 t)) = f(t) where q = exp(2 Pi i t). - Michael Somos, Dec 11 2008
a(n) ~ exp(2*Pi*sqrt(2*n/17)) / (2^(3/4) * 17^(1/4) * n^(3/4)). - Vaclav Kotesovec, Jun 28 2018
EXAMPLE
T34a = 1/q + q + 3*q^3 + 4*q^5 + 6*q^7 + 7*q^9 + 13*q^11 + 16*q^13 + ...
MATHEMATICA
QP = QPochhammer; s = QP[q^4]^2*(QP[q^34]^5/(QP[q]*QP[q^2]*QP[q^17]^3* QP[q^68]^2))-q^4*QP[q^2]^5*(QP[q^68]^2/(QP[q]^3*QP[q^4]^2*QP[q^17]* QP[q^34]))+O[q]^50; CoefficientList[s, q] (* Jean-François Alcover, Nov 13 2015, adapted from PARI *)
nmax = 60; CoefficientList[Series[Product[(1+x^k) * (1+x^(2*k))^2 * (1+x^(17*k)) * (1+x^(34*k-17))^2, {k, 1, nmax}] - x^4*Product[(1+x^k) * (1+x^(2*k-1))^2 * (1+x^(17*k)) * (1+x^(34*k))^2, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, May 01 2017 *)
PROG
(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^4 + A)^2 * eta(x^34 + A)^5 / (eta(x + A) * eta(x^2 + A) * eta(x^17 + A)^3 * eta(x^68 + A)^2) - x^4 * eta(x^2 + A)^5 * eta(x^68 + A)^2 / (eta(x + A)^3 * eta(x^4 + A)^2 * eta(x^17 + A) * eta(x^34 + A)), n))} /* Michael Somos, Dec 15 2008 */
CROSSREFS
KEYWORD
nonn
AUTHOR
N. J. A. Sloane, Nov 27 2000
EXTENSIONS
Erroneous zero at start of sequence removed by Michael Somos, Sep 30 2009
STATUS
approved