login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A113430 Expansion of f(-x, -x^2) * f(-x^10, -x^20) / f(-x^2, -x^8) in powers of x where f(, ) is Ramanujan's general theta function. 4
1, -1, 0, -1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 (list; graph; refs; listen; history; text; internal format)
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^4, b = x.

LINKS

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

M. 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^7, x^8) - x * f(x^2, x^13) in power of x.

Expansion of G(x^2) * f(-x) where G() is the g.f. of A003114.

Euler transform of period 10 sequence [ -1, 0, -1, -1, -1, -1, -1, 0, -1, -1, ...].

|a(n)| is the characteristic function of the numbers in A093722.

The exponents in the q-series q * A(q^120) are the square of the numbers in A057538.

G.f.: Prod_{k>0} (1 - x^k) / ((1 - x^(10*k - 2)) * (1 - x^(10*k - 8))) = Sum_{k in Z} x^((15*k^2 + k) / 2) - x^((15*k^2 - 11*k + 2) / 2).

EXAMPLE

G.f. = 1 - x - x^3 + x^7 + x^8 - x^14 - x^20 + x^29 + x^31 - x^42 - x^52 + ...

G.f. = q - q^121 - q^361 + q^841 + q^961 - q^1681 - q^2401 + q^3481 + q^3721 + ...

MATHEMATICA

a[ n_] := SeriesCoefficient[ QPochhammer[ x] / (QPochhammer[ x^2, x^10] QPochhammer[ x^8, x^10]), {x, 0, n}]; (* Michael Somos, Jan 06 2016 *)

PROG

(PARI) {a(n) = my(m); if( n<0 || !issquare( n*120 + 1, &m) || 1!=gcd(m, 30), 0, (-1)^(m%30\10))};

(PARI)  {a(n) = if( n<0, 0, polcoeff( prod( k=1, n, 1 - x^k * [1, 1, 0, 1, 1, 1, 1, 1, 0, 1][k%10 + 1], 1 + x * O(x^n)), n))};

CROSSREFS

Cf. A003114, A010815, A057538, A093722.

Sequence in context: A207710 A207735 A208546 * A214529 A113681 A295895

Adjacent sequences:  A113427 A113428 A113429 * A113431 A113432 A113433

KEYWORD

sign

AUTHOR

Michael Somos, Oct 31 2005

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified August 22 02:44 EDT 2019. Contains 326169 sequences. (Running on oeis4.)