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

0,1

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

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

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

Example

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 + ...

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 *)

Prog

(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))};

Crossrefs

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

Sequence in context: A267870 A267878 A231367 * A133101 A258769 A266377

Adjacent sequences: A113425 A113426 A113427 * A113429 A113430 A113431

Keyword

sign,easy

Author

Michael Somos, Oct 31 2005

Status

approved