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 phi(x^3) * chi(-x)^2 / f(x^4, x^8) in powers of x where phi(), chi(), f() are Ramanujan theta functions.
Expansion of f(-x, -x^5)^2 / (f(x^4, x^8) * f(x^6, x^18)) in powers of x where f(, ) is Ramanujan's general theta functions.
Expansion of q^(1/4) * eta(q)^2 * eta(q^4) * eta(q^6)^5 * eta(q^24) / (eta(q^2)^2 * eta(q^3)^2 *eta(q^8) * eta(q^12)^4) in powers of q.
Euler transform of period 24 sequence [-2, 0, 0, -1, -2, -3, -2, 0, 0, 0, -2, 0, -2, 0, 0, 0, -2, -3, -2, -1, 0, 0, -2, 0, ...].
G.f.: Product_{k>0} (1 - x^k + x^(2*k))^2 * (1 - x^(4*k) + x^(8*k)) / (1 + x^(6*k))^3.
a(n) = A134178(2*n + 1). a(6*n + 3) = a(6*n + 5) = 0.
EXAMPLE
G.f. = 1 - 2*x + x^2 - x^4 + 2*x^7 + x^8 - x^12 - 4*x^13 - x^14 + 2*x^16 + ...
G.f. = q^-1 - 2*q^3 + q^7 - q^15 + 2*q^27 + q^31 - q^47 - 4*q^51 - q^55 + ...
MATHEMATICA
a[ n_] := SeriesCoefficient[ QPochhammer[ x, x^2]^2 QPochhammer[ -x^3, x^6]^2 QPochhammer[ x^4, x^8] QPochhammer[ x^6, -x^6], {x, 0, n}];
PROG
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^2 * eta(x^4 + A) * eta(x^6 + A)^5 * eta(x^24 + A) / (eta(x^2 + A)^2 * eta(x^3 + A)^2 * eta(x^8 + A) * eta(x^12 + A)^4), n))};
(PARI) lista(nn) = {q='q+O('q^nn); Vec(eta(q)^2*eta(q^4)*eta(q^6)^5*eta(q^24)/(eta(q^2)^2*eta(q^3)^2 *eta(q^8)*eta(q^12)^4))} \\ Altug Alkan, Mar 21 2018
CROSSREFS
KEYWORD
sign
AUTHOR
Michael Somos, Feb 26 2017
STATUS
approved