A121373 Expansion of f(x) = f(x, -x^2) in powers of x where f(, ) is Ramanujan's general theta function.

1, 1, -1, 0, 0, -1, 0, -1, 0, 0, 0, 0, -1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0

Offset: 0

Author

Michael Somos, Jul 24 2006

Keywords

• sign

Comments

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

This is an example of the quintuple product identity in the form f(a*b^4, a^2/b) - (a/b) * f(a^4*b, b^2/a) = f(-a*b, -a^2*b^2) * f(-a/b, -b^2) / f(a, b) where a = -x^3, b = -x. - Michael Somos, Jul 11 2012

Number 5 of the 14 primitive eta-products which are holomorphic modular forms of weight 1/2 listed by D. Zagier on page 30 of "The 1-2-3 of Modular Forms". - Michael Somos, May 04 2016

Examples

			G.f. = 1 + x - x^2 - x^5 - x^7 - x^12 + x^15 + x^22 + x^26 + x^35 + ...
G.f. = q + q^25 - q^49 - q^121 - q^169 - q^289 + q^361 + q^529 + ...
		

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
• Eric Weisstein's World of Mathematics, Quintuple Product Identity

Crossrefs

Cf. A010815, A080995, A247133, A247223.

Programs

Mathematica

a[ n_] := SeriesCoefficient[ Product[ 1 - (-x)^k, {k, n}], {x, 0, n}]; (* Michael Somos, Nov 14 2011 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ -x], {x, 0, n}]; (* Michael Somos, Jul 06 2013 *)
a[ n_] := SeriesCoefficient[ (EllipticTheta[ 1, Pi/12, x^4] + EllipticTheta[ 2, Pi/12, x^4]) / Sqrt[6], {x, 0, 24 n + 1}] // Simplify; (* Michael Somos, Mar 20 2015 *)

PARI

{a(n) = if( issquare( 24*n + 1, &n), kronecker( 6, n))};

PARI

{a(n) = if( n<0, 0, polcoeff( eta( -x + x * O(x^n)), n))};

Formula

Expansion of q^(-1/4) * (theta_1( Pi/12, q) + theta_2( Pi/12, q)) / sqrt(6) in powers of q^6. - Michael Somos, Jul 06 2013

Expansion of q^(-1/24) * eta(q^2)^3 / (eta(q) * eta(q^4)) in powers of q.

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

a(n) = b(24*n + 1) where b() is multiplicative with b(p^2e) = (-1)^e if p == 7, 11, 13, 17 (mod 24), b(p^2e) = +1 if p == 1, 5, 19, 23 (mod 24) and b(p^(2e-1)) = b(2^e) = b(3^e) = 0 if e>0.

G.f.: (1 + x) * (1 - x^2) * (1 + x^3) * (1 - x^4) * ...

G.f.: 1 + x - x^2*(1 + x) + x^3*(1 + x)*(1 - x^2) - x^4*(1 + x)*(1 - x^2)*(1 + x^3) + ...

a(5*n + 3) = a(5*n + 4) = 0. a(25*n + 1) = a(n).

G.f.: Sum_{k>=0} a(k) * x^(24*k + 1) = Sum_{k in Z} (-1)^floor((k+1)/2) * x^(6*k + 1)^2.

a(n) = (-1)^n * A010815(n). |a(n)| = A080995(n).

Expansion of f(-x^5, -x^7) + x * f(-x, -x^11) in powers of x. - Michael Somos, Jan 10 2015

G.f. is a period 1 Fourier series which satisfies f(-1 / (2304 t)) = 48^(1/2) (t/i)^(1/2) f(t) where q = exp(2 Pi i t). - Michael Somos, May 05 2016

G.f.: exp(Sum_{k>=1} (-1)^(k+1)*x^k/(k*(1 - (-x)^k))). - Ilya Gutkovskiy, Jun 08 2018