A224877 Expansion of f(x^29, x^31) - x^3 * f(x^11, x^49) in powers of x where f() is a Ramanujan theta function.

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

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^17, b = x^3.

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

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^7) * f(x^7, x^13) / f(-x^10) = f(-x^10) * f(-x^6, -x^14) / f(x^3, x^17) in powers of x where f() is a Ramanujan theta function.

Euler transform of period 40 sequence [ 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, -1, 0, 0, -1, 0, 0, -1, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, -1, ...].

G.f.: Sum_{k in Z} x^(k * (30*k + 10)) * (x^(-9*k) - x^(9*k + 3)).

a(7*n + 2) = a(7*n + 4) = a(7*n + 5) = a(11*n + 2) = a(11*n + 4) = a(11*n + 5) = a(11*n + 6) = a(11*n + 10) = 0.

Example

G.f. = 1 - x^3 - x^14 + x^29 + x^31 - x^52 - x^85 + x^118 + x^122 - x^161 - x^216 + ...
G.f. = q - q^361 - q^1681 + q^3481 + q^3721 - q^6241 - q^10201 + q^14161 + q^14641 + ...

Mathematica

a[ n_] := SeriesCoefficient[ QPochhammer[ x^3, x^10] QPochhammer[ x^7, x^10] QPochhammer[ -x^7, x^20] QPochhammer[ -x^13, x^20] QPochhammer[ x^20], {x, 0, n}];

Prog

(PARI) {a(n) = local(m); if( n<0 || !issquare( 120*n + 1, &m), 0, if( m%20 > 9, m = -m); if( m%20 != 1, 0, 1 - (m\20)%3))};

Crossrefs

Sequence in context: A154281 A154282 A359552 * A178600 A359170 A014702

Adjacent sequences: A224874 A224875 A224876 * A224878 A224879 A224880

Keyword

sign

Author

Michael Somos, Jul 23 2013

Status

approved