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A216046 Expansion of (chi(-x) / chi^3(-x^3))^2 in powers of x where chi() is a Ramanujan theta function. 6
1, -2, 1, 4, -8, 2, 14, -24, 6, 38, -63, 16, 92, -150, 36, 208, -329, 78, 440, -684, 160, 884, -1358, 312, 1710, -2592, 590, 3196, -4796, 1082, 5800, -8632, 1929, 10270, -15162, 3364, 17784, -26078, 5750, 30192, -44010, 9644, 50369, -73012, 15916, 82698 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

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

Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).

LINKS

G. C. Greubel, Table of n, a(n) for n = 0..1000

M. Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

FORMULA

Expansion of q^(-2/3) * (c(q^2) / c(q))^2 in powers of a where c() is a cubic AGM theta function (see A005882)

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

Euler transform of period 6 sequence [ -2, 0, 4, 0, -2, 0, ...].

Given g.f. A(x), then B(x) = x^2 * A(x^3) satisfies 0 = f(B(x), B(x^3)) where f(u, v) = u^3 * (6*v * (1 + 2*v))^2 - ((v+v^2+v^3) - u^3 * (1 + 4*v + 16*v^2))^2.

Given g.f. A(x), then B(x) = x^2 * A(x^3) satisfies 0 = f(B(x), B(x^2), B(x^4)) where f(u, v, w) = v * (u + 2*w) * (v + 2*u*w) - u*w * (1 + 8*v^3).

G.f. is a period 1 Fourier series which satisfies f(-1 / (18 t)) = (1/4) g(t) where q = exp(2 Pi i t) and g() is g.f. for A242405. - Michael Somos, May 13 2014

G.f.: Product_{k>0} ((1 - x^(2*k - 1)) / (1 - x^(6*k - 3))^3)^2.

a(n) = (-1)^n * A164614(n) = A128111(2*n + 1) = -A092848(2*n + 1) = -A182032(12*n+8).

Convolution square of A092848.

EXAMPLE

G.f. = 1 - 2*x + x^2 + 4*x^3 - 8*x^4 + 2*x^5 + 14*x^6 - 24*x^7 + 6*x^8 + 38*x^9 + ...

G.f. = q^2 - 2*q^5 + q^8 + 4*q^11 - 8*q^14 + 2*q^17 + 14*q^20 - 24*q^23 + 6*q^26 + ...

MATHEMATICA

a[ n_] := SeriesCoefficient[ Product[ 1 - x^k, {k, 1, n, 2}]^2 / Product[ 1 - x^k, {k, 3, n, 6}]^6, {x, 0, n}]; (* Michael Somos, Dec 03 2013 *)

a[ n_] := SeriesCoefficient[ (QPochhammer[ x, x^2] / QPochhammer[ x^3, x^6]^3)^2, {x, 0, n}]; (* Michael Somos, May 13 2014 *)

PROG

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

CROSSREFS

Cf. A092848, A128111, A164614, A182032, A242405.

Sequence in context: A275364 A160323 A128411 * A164614 A254102 A094511

Adjacent sequences:  A216043 A216044 A216045 * A216047 A216048 A216049

KEYWORD

sign

AUTHOR

Michael Somos, Aug 31 2012

STATUS

approved

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Last modified February 27 15:39 EST 2020. Contains 332307 sequences. (Running on oeis4.)