OFFSET
0,7
COMMENTS
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1500
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
FORMULA
Expansion of q^(1/6) * eta(q) * eta(q^6)^2 * eta(q^9) / (eta(q^2) * eta(q^3)^2 * eta(q^18)) in powers of q.
Euler transform of period 18 sequence [ -1, 0, 1, 0, -1, 0, -1, 0, 0, 0, -1, 0, -1, 0, 1, 0, -1, 0, ...].
Given g.f. A(x) then B(q) = A(q^6) / q satisfies 0 = f(B(q), B(q^2), B(q^4) ) where f(u, v, w) = (u^2 + v) * v + (u^2 - v) * w^2.
Given g.f. A(x) then B(q) = A(q^3)^2 / q satisfies 0 = f(B(q), B(q^2)) where f(u, v) = (u - v^2) * (u^2 - v) * (1 + u * v) - (2 * u * v)^2.
G.f. is a period 1 Fourier series which satisfies f(-1 / (648 t)) = g(t) where q = exp(2 Pi i t) and g() is the g.f. for A134132.
G.f.: ( Product_{k>0} (1 + x^k) * (1 + x^(9*k)) / (1 + x^(3*k))^2 )^(-1).
EXAMPLE
G.f. = 1 - x + x^3 - x^4 - x^5 + 2*x^6 - 2*x^7 + 2*x^9 - 2*x^10 - x^11 + 4*x^12 + ...
G.f. = 1/q - q^5 + q^17 - q^23 - q^29 + 2*q^35 - 2*q^41 + 2*q^53 - 2*q^59 - q^65 + ...
MATHEMATICA
a[ n_] := SeriesCoefficient[ QPochhammer[ x, x^2] QPochhammer[ -x^3, x^3]^2 QPochhammer[ x^9, x^18] , {x, 0, n}]; (* Michael Somos, Oct 27 2015 *)
eta[q_] := q^(1/24)*QPochhammer[q]; b := q^(1/6)*eta[q]*eta[q^6]^2* eta[q^9]/(eta[q^2]*eta[q^3]^2*eta[q^18]); a := CoefficientList[ Series[ b, {q, 0, 80}], q]; Table[a[[n]], {n, 1, 80}] (* G. C. Greubel, Jul 03 2018 *)
PROG
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^6 + A)^2 * eta(x^9 + A) / (eta(x^2 + A) * eta(x^3 + A)^2 * eta(x^18 + A)), n))};
CROSSREFS
KEYWORD
sign
AUTHOR
Michael Somos, Oct 10 2007
STATUS
approved