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A134132 Expansion of chi(-x^3)^2 / (chi(-x) * chi(-x^9)) in power of x where chi() is a Ramanujan theta function. 2
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified May 7 06:57 EDT 2021. Contains 343636 sequences. (Running on oeis4.)