A263456 Expansion of phi(-q) * f(-q^8)^3 / f(-q^24) in powers of q where phi(), f() are Ramanujan theta functions.

1, -2, 0, 0, 2, 0, 0, 0, -3, 4, 0, 0, -6, 0, 0, 0, 2, 6, 0, 0, 0, 0, 0, 0, 0, -14, 0, 0, 12, 0, 0, 0, -3, 0, 0, 0, -4, 0, 0, 0, 12, 6, 0, 0, -6, 0, 0, 0, -6, -14, 0, 0, 0, 0, 0, 0, -6, 24, 0, 0, 0, 0, 0, 0, 2, 12, 0, 0, -6, 0, 0, 0, -12, -24, 0, 0, 12, 0, 0, 0

Offset: 0

Author

Michael Somos, Oct 19 2015

Keywords

• sign

Comments

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

Examples

			G.f. = 1 - 2*x + 2*x^4 - 3*x^8 + 4*x^9 - 6*x^12 + 2*x^16 + 6*x^17 - 14*x^25 + ...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..2500
• Michael Somos, Introduction to Ramanujan theta functions
• Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Crossrefs

Cf. A263452.

Programs

Mathematica

a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, q] QPochhammer[ q^8]^3 / QPochhammer[ q^24], {q, 0, n}];

PARI

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^2 * eta(x^8 + A)^3 / (eta(x^2 + A) * eta(x^24 + A)), n))};

Formula

Expansion of eta(q)^2 * eta(q^8)^3 / (eta(q^2) * eta(q^24)) in powers of q.

Euler transform of period 24 sequence [ -2, -1, -2, -1, -2, -1, -2, -4, -2, -1, -2, -1, -2, -1, -2, -4, -2, -1, -2, -1, -2, -1, -2, -3, ...].

G.f. is a period 1 Fourier series which satisfies f(-1 / (144 t)) = 279936^(1/2) (t/I)^(3/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A263452.

a(4*n + 2) = a(4*n + 3) = a(8*n + 5) = a(9*n + 6) = 0.