A113428 Expansion of f(-x^2, -x^3) in powers of x where f(, ) is Ramanujan's general theta function.

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

Offset: 0

Author

Michael Somos, Oct 31 2005

Keywords

• sign
• easy

Comments

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

See the Hardy-Wright reference for the identity given as g.f. formula below. - Wolfdieter Lang, Oct 28 2016

Examples

			G.f. = 1 - x^2 - x^3 + x^9 + x^11 - x^21 - x^24 + x^38 + x^42 - x^60 - x^65 + ...
G.f. = q - q^81 - q^121 + q^361 + q^441 - q^841 - q^961 + q^1521 + q^1681 + ...
		

References

• G. H. Hardy, Ramanujan, AMS Chelsea Publ., Providence, RI, 2002, p. 93.
• G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, Fifth ed., Clarendon Press, Oxford, 2003, Theorem 356, p. 284.

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..10000
• Michael Somos, Introduction to Ramanujan theta functions
• Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
• Eric Weisstein's World of Mathematics, Rogers-Ramanujan Identities

Crossrefs

Cf. A003114, A057569, A090771, A010815, A113429.

Programs

Mathematica

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

PARI

{a(n) = if( n<0, 0, polcoeff( prod(k=1, n, 1 - x^k * [1, 0, 1, 1, 0][k%5 + 1], 1 + x * O(x^n)), n))};

Formula

Expansion of G(x) * f(-x) in powers of x where G() is the g.f. of A003114.

Euler transform of period 5 sequence [ 0, -1, -1, 0, -1, ...].

|a(n)| is the characteristic function of the numbers in A057569.

The exponents in the q-series q * A(q^40) are the square of the numbers in A090771.

G.f.: Sum_{k in Z} (-1)^k * x^((5*k^2 + k)/2) = Prod_{k>0} (1 - x^(5*k)) * (1 - x^(5*k - 2)) * (1 - x^(5*k - 3)).

Convolution of A003114 and A010815.

From Wolfdieter Lang, Oct 30 2016: (Start)

a(n) = (-1)^k if n = b(2*k+1) for k >= 0, a(n) = (-1)^k if n = b(2*k), for k >= 1, and a(n) = 0 otherwise, where b(n) = A057569(n). See the third formula.

G.f.: Sum_{n>=0} (-1)^n*x^(n*(5*n+1)/2)*(1-x^(2*(2*n+1))). See the Hardy reference, p. 93, eq. (6.11.1) with k=2, a=x and C_n = 1.

(End)

G.f.: Sum_{n >= 0} x^(n^2)*Product_{k >= n+1} 1 - x^k. Cf. A113429. - Peter Bala, Feb 12 2021