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 q^(-1/3) * (eta(q)^11 / eta(q^3) + 27 * eta(q^3)^11 / eta(q)) in powers of q.
a(n) = b(3*n + 1) where b(n) is multiplicative and b(3^e) = 0^e, b(p^e) = (1 + (-1)^e)/2 * p^(2*e) if p == 2 (mod 3), b(p^e) = b(p) * b(p^(e-1)) - p^4 * b(p^(e-2)) if p == 1 (mod 3) where b(p) = (y^2 - 2*p)^2 - 2*p^2, 4*p = y^2 + 27*x^2.
G.f. is a period 1 Fourier series which satisfies f(-1 / (27 t)) = 27^(5/2) (t/i)^5 f(t) where q = exp(2 Pi i t).
a(4*n + 3) = 0.
EXAMPLE
G.f. = 1 + 16*x + 71*x^2 - 337*x^4 + 256*x^5 - 601*x^6 + 625*x^8 + ...
G.f. = q + 16*q^4 + 71*q^7 - 337*q^13 + 256*q^16 - 601*q^19 + ...
MATHEMATICA
a[ n_] := SeriesCoefficient[ QPochhammer[ x]^11 / QPochhammer[ x^3] + 27 x QPochhammer[ x^3]^11 / QPochhammer[ x], {x, 0, n}];
PROG
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^11 / eta(x^3 + A) + 27 * x * eta(x^3 + A)^11 / eta(x + A), n))};
(PARI) {a(n) = my(A, p, e, x, y, a0, a1); n = 3*n + 1; if( n<1, 0, A = factor(n); prod( k=1, matsize(A)[1], [p, e] = A[k, ]; if( p==3, 0, p%3==2, if( e%2, 0, p^(2*e)), for( x=1, sqrtint(4*p\27), if( issquare(4*p - 27*x^2, &y), break)); y = (y^2 - 2*p)^2 - 2*p^2; a0=1; a1=y; for( i=2, e, x = y*a1 - p^4*a0; a0=a1; a1=x); a1)))};
CROSSREFS
KEYWORD
sign
AUTHOR
Michael Somos, Jun 08 2015
STATUS
approved