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^(-2/3) * (c(q^2) / c(q))^2 in powers of a where c() is a cubic AGM theta function (see A005882)
Expansion of q^(-2/3) * (eta(q) * eta(q^6)^3 / (eta(q^2) * eta(q^3)^3))^2 in powers of q.
Euler transform of period 6 sequence [ -2, 0, 4, 0, -2, 0, ...].
Given g.f. A(x), then B(x) = x^2 * A(x^3) satisfies 0 = f(B(x), B(x^3)) where f(u, v) = u^3 * (6*v * (1 + 2*v))^2 - ((v+v^2+v^3) - u^3 * (1 + 4*v + 16*v^2))^2.
Given g.f. A(x), then B(x) = x^2 * A(x^3) satisfies 0 = f(B(x), B(x^2), B(x^4)) where f(u, v, w) = v * (u + 2*w) * (v + 2*u*w) - u*w * (1 + 8*v^3).
G.f. is a period 1 Fourier series which satisfies f(-1 / (18 t)) = (1/4) g(t) where q = exp(2 Pi i t) and g() is g.f. for A242405. - Michael Somos, May 13 2014
G.f.: Product_{k>0} ((1 - x^(2*k - 1)) / (1 - x^(6*k - 3))^3)^2.
Convolution square of A092848.
EXAMPLE
G.f. = 1 - 2*x + x^2 + 4*x^3 - 8*x^4 + 2*x^5 + 14*x^6 - 24*x^7 + 6*x^8 + 38*x^9 + ...
G.f. = q^2 - 2*q^5 + q^8 + 4*q^11 - 8*q^14 + 2*q^17 + 14*q^20 - 24*q^23 + 6*q^26 + ...
MATHEMATICA
a[ n_] := SeriesCoefficient[ Product[ 1 - x^k, {k, 1, n, 2}]^2 / Product[ 1 - x^k, {k, 3, n, 6}]^6, {x, 0, n}]; (* Michael Somos, Dec 03 2013 *)
a[ n_] := SeriesCoefficient[ (QPochhammer[ x, x^2] / QPochhammer[ x^3, x^6]^3)^2, {x, 0, n}]; (* Michael Somos, May 13 2014 *)
PROG
(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x + A) * eta(x^6 + A)^3 / (eta(x^2 + A) * eta(x^3 + A)^3))^2, n))};
CROSSREFS
KEYWORD
sign
AUTHOR
Michael Somos, Aug 31 2012
STATUS
approved