OFFSET
0,2
COMMENTS
LINKS
G. C. Greubel, 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
FORMULA
Expansion of f(-x)^4 + 7 * x * f(-x^7)^4 = chi(-x) * chi(-x^7) * (psi(x)^4 + 7 * x^3 * psi(x^7)^4) in powers of x where psi(), chi(), f() are Ramanujan theta functions.
Expansion of (phi(-x)^4 + 7 * phi(-x^7)^4) / (8 * chi(-x) * chi(-x^7)) in powers of x^2 where phi(), chi(), f() are Ramanujan theta functions.
a(n) = b(6*n + 1) where b(n) is multiplicative and b(2^e) = b(3^e) = 0^e, b(p^e) = (-p)^(e/2) (1 + (-1)^e)/2 if p == 5 (mod 6).
G.f. is a period 1 Fourier series which satisfies f(-1 / (252 t)) = 252 (t/i)^2 * f(t) where q = exp(2 Pi i t).
EXAMPLE
G.f. = 1 + 3*x + 2*x^2 + 8*x^3 - 5*x^4 - 4*x^5 - 10*x^6 + 8*x^7 - 19*x^8 + ...
G.f. = q + 3*q^7 + 2*q^13 + 8*q^19 - 5*q^25 - 4*q^31 - 10*q^37 + 8*q^43 + ...
MATHEMATICA
a[ n_] := SeriesCoefficient[ QPochhammer[ x]^4 + 7 x QPochhammer[ x^7]^4, {x, 0, n}]; (* Michael Somos, Sep 02 2015 *)
PROG
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^4 + 7 * x * eta(x^7 + A)^4, n))};
CROSSREFS
KEYWORD
sign
AUTHOR
Michael Somos, Feb 07 2010
STATUS
approved