A137569 Expansion of f(-x) / f(-x^3) in powers of x where f() is a Ramanujan theta function.

1, -1, -1, 1, -1, 0, 2, -1, -1, 3, -2, -1, 4, -3, -2, 5, -4, -2, 8, -6, -4, 10, -7, -4, 14, -10, -6, 18, -13, -7, 24, -17, -10, 30, -21, -12, 40, -28, -17, 49, -35, -19, 63, -44, -26, 78, -55, -31, 98, -69, -40, 120, -84, -47, 150, -105, -61, 182, -127, -71

Offset: 0

Author

Michael Somos, Jan 26 2008

Keywords

• sign

Comments

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

Examples

			G.f. = 1 - x - x^2 + x^3 - x^4 + 2*x^6 - x^7 - x^8 + 3*x^9 - 2*x^10 - x^11 + ...
G.f. = 1/q - q^11 - q^23 + q^35 - q^47 + 2*q^71 - q^83 - q^95 + 3*q^107 + ...
		

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..10000
• Michael Somos, Introduction to Ramanujan theta functions
• Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Crossrefs

Cf. A000726, A035940, A035941, A035943, A058095, A112157, A263050, A263051.

Programs

Mathematica

a[ n_] := SeriesCoefficient[ QPochhammer[ x] / QPochhammer[ x^3], {x, 0, n}]; (* Michael Somos, Oct 08 2015 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ x, x^3] QPochhammer[ x^2, x^3], {x, 0, n}]; (* Michael Somos, Oct 08 2015 *)

PARI

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

Formula

Expansion of q^(1/12) eta(q) / eta(q^3) in powers of q.

Euler transform of period 3 sequence [ -1, -1, 0, ...].

Given g.f. A(x) then B(q) = A(q^6)^2 / q satisfies 0 = f(B(q), B(q^2), B(q^4)) where f(u, v, w) = 4*v^2 + (u^2 - v) * (w^2 + v).

G.f. is a period 1 Fourier series which satisfies f(-1 / (432 t)) = 3^(1/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A000726.

G.f.: Product_{k>0} (1 - x^(3*k-1)) * (1 - x^(3*k-2)).

a(3*n) = A035943(n). a(3*n + 1) = - A035941(n). a(3*n + 2) = - A035940(n).

Convolution inverse of A000726.

Convolution square is A112157. Convolution 4th power is A058095. - Michael Somos, Oct 08 2015

a(2*n) = A263050(n). a(2*n + 1) = - A263051(n). - Michael Somos, Oct 08 2015

G.f.: (Product_{k>0} (1 + x^k + x^(2*k)))^-1. - Michael Somos, Oct 08 2015

a(n) = -(1/n)*Sum_{k=1..n} A046913(k)*a(n-k), a(0) = 1. - Seiichi Manyama, Mar 25 2017