OFFSET
0,1
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
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
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
EXAMPLE
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 + ...
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 *)
PROG
(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))};
CROSSREFS
KEYWORD
sign
AUTHOR
Michael Somos, Oct 22 2012
STATUS
approved