A257657 Expansion of f(-x, -x) * f(-x^6, -x^6) / f(x, x^2) in powers of x where f(,) is Ramanujan's general theta function.

1, -3, 2, 1, -1, -1, 3, -1, 0, -2, -2, 2, 1, -3, 3, 4, -1, -3, 1, 0, -1, -2, 0, 3, 1, -6, 2, 4, -4, -1, 4, 2, -1, -3, 0, 5, -1, -9, 5, 7, -4, -7, 4, 5, -3, -4, 0, 8, -1, -13, 4, 11, -7, -7, 7, 6, -1, -10, 0, 14, -1, -15, 8, 15, -10, -14, 8, 11, -7, -13, 2, 17

Offset: 0

Author

Michael Somos, Jul 26 2015

Keywords

• sign

Comments

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

Examples

			G.f. = 1 - 3*x + 2*x^2 + x^3 - x^4 - x^5 + 3*x^6 - x^7 - 2*x^9 - 2*x^10 + ...
G.f. = 1/q - 3*q^23 + 2*q^47 + q^71 - q^95 - q^119 + 3*q^143 - q^167 + ...
		

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. A053250, A260413.

Programs

Mathematica

a[ n_] := SeriesCoefficient[ QPochhammer[ x, x^2]^2 QPochhammer[ x] QPochhammer[ -x^3] / QPochhammer[ x^3], {x, 0, n}];

PARI

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^3 * eta(x^6 + A)^3 / (eta(x^2 + A)^2 * eta(x^3 + A)^2 * eta(x^12 + A)), n))};

Formula

Expansion of q^(1/24) * eta(q)^3 * eta(q^6)^3 / (eta(q^2)^2 * eta(q^3)^2 * eta(q^12)) in powers of q.

Euler transform of period 12 sequence [ -3, -1, -1, -1, -3, -2, -3, -1, -1, -1, -3, -1, ...].

a(n) = 2 * A260413(n) - A053250(n).