A204220 Expansion of f(-x^2, -x^3) * f(-x^2, -x^4) / f(-x, -x^2) in powers of x where f(, ) is Ramanujan's general theta function.

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

Offset: 0

Author

Michael Somos, Jan 13 2012

Keywords

• sign

Comments

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

Examples

			G.f. = 1 + x - x^8 - x^13 - x^17 - x^24 + x^45 + x^56 + x^64 + x^77 - x^112 + ...
G.f. = q + q^16 - q^121 - q^196 - q^256 - q^361 + q^676 + q^841 + q^961 + q^1156 + ...
		

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

Crossrefs

Cf. A003114, A010815.

Programs

Mathematica

a[ n_] := SeriesCoefficient[ QPochhammer[ x^2, x^5] QPochhammer[ x^3, x^5] QPochhammer[ x^5] QPochhammer[ -x, x], {x, 0, n}]; (* Michael Somos, Jan 06 2016 *)
a[ n_] := SeriesCoefficient[ Product[ (1 - x^k)^{ -1, 1, 0, 0, 0, 0, 0, 1, -1, 1}[[Mod[k, 10, 1]]], {k, n}], {x, 0, n}]; (* Michael Somos, Jan 06 2016 *)

PARI

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

Formula

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

Expansion of f(-x^13, -x^17) + x * f(-x^7, -x^23) in powers of x.

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

G.f.: Sum_{k} (-1)^k * x^(15*k^2) * (x^(2*k) + x^(8*k + 1)) = Product_{k>0} (1 - x^(10*k)) * (1 - x^(10*k -2)) * (1 - x^(10*k -8)) / ((1 - x^(10*k -1)) * (1 - x^(10*k -9))).

|a(n)| is the characteristic function of A204221.

The exponents in the q-series q * A(q^15) are the squares of the numbers == +- 1 or +- 4 (mod 15).