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A259657
Expansion of phi(-x^3) * f(-x^4)^3 / f(-x^12) in powers of x where phi(), f() are Ramanujan theta functions.
1
1, 0, 0, -2, -3, 0, 0, 6, 0, 0, 0, 0, 8, 0, 0, -12, -9, 0, 0, 6, 0, 0, 0, 0, 12, 0, 0, -2, -12, 0, 0, 18, 0, 0, 0, 0, 6, 0, 0, -24, -12, 0, 0, 6, 0, 0, 0, 0, 20, 0, 0, -12, -12, 0, 0, 24, 0, 0, 0, 0, 24, 0, 0, -12, -21, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0, -14, -24
OFFSET
0,4
COMMENTS
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).
LINKS
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
FORMULA
Expansion of phi(-x^3) * b(x^4) in powers of x where phi() is a Ramanujan theta function and b() is a cubic AGM theta function.
Expansion of eta(q^3)^2 * eta(q^4)^3 / (eta(q^6) * eta(q^12)) in powers of q.
Euler transform of period 12 sequence [ 0, 0, -2, -3, 0, -1, 0, -3, -2, 0, 0, -3, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (72 t)) = 72^(3/2) (t/i)^(3/2) g(t) where q = exp(2 Pi i t) and g(t) is the g.f. for A259655.
G.f.: Product_{k>0} (1 - x^(4*k))^3 / ((1 + x^(3*k))^2 * (1 + x^(6*k))).
a(3*n + 1) = -3 * A143161(n-1). a(3*n + 2) = a(4*n + 1) = a(4*n + 2) = 0. a(12*n) = A014453(n).
EXAMPLE
G.f. = 1 - 2*x^3 - 3*x^4 + 6*x^7 + 8*x^12 - 12*x^15 - 9*x^16 + 6*x^19 + ...
MATHEMATICA
a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, x^3] QPochhammer[ x^4]^3 / QPochhammer[ x^12], {x, 0, n}];
a[ n_] := SeriesCoefficient[ QPochhammer[x^4]^3 / (QPochhammer[ -x^3, x^3]^2 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^3 + A)^2 * eta(x^4 + A)^3 / (eta(x^6 + A) * eta(x^12 + A)), n))};
CROSSREFS
KEYWORD
sign
AUTHOR
Michael Somos, Jul 02 2015
STATUS
approved