A218171 Expansion of f(x^11, x^13) - x * f(x^5, x^19) in powers of x where f(, ) is Ramanujan's general theta function.

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

Offset: 0

Author

Michael Somos, Oct 22 2012

Keywords

• sign
• changed

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^7, b = x. - Michael Somos, Nov 09 2014

This is exercise 0.13 3 on page 45 of Cooper [2017]. - Michael Somos, Aug 30 2025

Examples

			G.f. = 1 - x - x^6 + x^11 + x^13 - x^20 - x^35 + x^46 + x^50 - x^63 - x^88 + ...
G.f. = q - q^49 - q^289 + q^529 + q^625 - q^961 - q^1681 + q^2209 + q^2401 + ...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..10000
• Shaun Cooper, Ramanujan's Theta Functions, Springer International (2017).
• 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, A115671.

Programs

Mathematica

a[ n_] := If[ n < 0, 0, If[ OddQ[ DivisorSigma[ 0, 48 n + 1]], JacobiSymbol[ 6, Sqrt[48 n + 1]], 0]]; (* Michael Somos, Nov 09 2014 *)
a[ n_] := SeriesCoefficient[ (QPochhammer[ -q] + QPochhammer[ q]) / 2, {q, 0, 2 n}]; (* Michael Somos, Nov 09 2014 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ q] (QPochhammer[ q^2]^3 / QPochhammer[ q]^2/ QPochhammer[ q^4] + 1) / 2, {q, 0, 2 n}]; (* Michael Somos, Nov 09 2014 *)

PARI

{a(n) = my(m); if( issquare(48*n + 1, &m), kronecker(6, m), 0)};

PARI

{a(n) = my(m); if( n<0, 0, m = 2*n; polcoeff( eta(x + x * O(x^m)), m))};

Formula

Expansion of f(x^3, x^5) * chi(-x) in powers of x where f(, ) is Ramanujan's general theta function and chi() is a Ramanujan theta function.

G.f.: Sum_{k in Z} x^(12*k^2 + k) - x^(12*k^2 + 7*k + 1).

a(n) = A010815(2*n) for all n in Z.

G.f.: Product_{j>0} (1-x^(8*j-1)) * (1-x^(8*j-7)) * (1-x^(8*j)) * (1-x^(16*j-6)) * (1-x^(16*j-10)). [Cooper 2017] - Michael Somos, Aug 30 2025