A134132 Expansion of chi(-x^3)^2 / (chi(-x) * chi(-x^9)) in power of x where chi() is a Ramanujan theta function.

1, 1, 1, 0, 0, 1, 1, 2, 1, 1, 1, 2, 3, 3, 3, 2, 2, 3, 5, 6, 5, 4, 4, 6, 9, 10, 9, 8, 8, 11, 14, 16, 15, 13, 14, 18, 24, 26, 25, 22, 23, 29, 36, 40, 38, 36, 38, 46, 56, 61, 60, 56, 59, 70, 84, 92, 90, 86, 90, 106, 125, 135, 134, 130, 136, 157, 181, 196, 195

Offset

0,8

Comments

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

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..2000

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

Expansion of q^(-1/6) * eta(q^2) * eta(q^3)^2 * eta(q^18) / (eta(q) * eta(q^6)^2 * eta(q^9)) in powers of q.

Euler transform of period 18 sequence [ 1, 0, -1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, -1, 0, 1, 0, ...].

Given g.f. A(x) then B(q) = A(q^6) * q satisfies 0 = f(B(q), B(q^2), B(q^4) ) where f(u, v, w) = (u^2 + v) * w^2 - (u^2 - v) * v.

Given g.f. A(x) then B(q) = A(q^3)^2 * q satisfies 0 = f(B(q), B(q^2)) where f(u, v) = (u - v^2) * (u^2 - v) * (1 + u * v) - (2 * u * v)^2.

G.f. is a period 1 Fourier series which satisfies f(-1 / (648 t)) = 1 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A134131.

G.f.: Product_{k>0} (1 + x^k) * (1 + x^(9*k)) / (1 + x^(3*k))^2.

-a(n) = A112178(3*n + 1). Convolution inverse of A134131.

a(n) ~ exp(2*Pi*sqrt(n/3)/3) / (2 * 3^(3/4) * n^(3/4)). - Vaclav Kotesovec, Sep 08 2015

Example

G.f. = 1 + x + x^2 + x^5 + x^6 + 2*x^7 + x^8 + x^9 + x^10 + 2*x^11 + 3*x^12 + ...
G.f. = q + q^7 + q^13 + q^31 + q^37 + 2*q^43 + q^49 + q^55 + q^61 + 2*q^67 + ...

Mathematica

nmax = 80; CoefficientList[Series[Product[(1 + x^k) * (1 + x^(9*k)) / (1 + x^(3*k))^2, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 08 2015 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ -x, x] QPochhammer[ x^3, x^6]^2 QPochhammer[ -x^9, x^9] , {x, 0, n}]; (* Michael Somos, Oct 27 2015 *)

Prog

(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A) * eta(x^3 + A)^2 * eta(x^18 + A) / (eta(x + A) * eta(x^6 + A)^2 * eta(x^9 + A)), n))};

Crossrefs

Cf. A112178, A134131.

Sequence in context: A114731 A035389 A129176 * A308121 A030424 A216656

Adjacent sequences: A134129 A134130 A134131 * A134133 A134134 A134135

Keyword

nonn

Author

Michael Somos, Oct 10 2007, Oct 21 2007

Status

approved