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A128128 Expansion of chi(-q^3) / chi^3(-q) in powers of q where chi() is a Ramanujan theta function. 7
1, 3, 6, 12, 21, 36, 60, 96, 150, 228, 342, 504, 732, 1050, 1488, 2088, 2901, 3996, 5460, 7404, 9972, 13344, 17748, 23472, 30876, 40413, 52644, 68268, 88152, 113364, 145224, 185352, 235734, 298800, 377514, 475488, 597108, 747690, 933672, 1162824 (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

Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015

M. Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

FORMULA

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

Euler transform of period 6 sequence [ 3, 0, 2, 0, 3, 0, ...].

G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = u^2 + v - 2*u*v^2.

G.f. A(x) satisfies 0 = f(A(x), A(x^3)) where f(u, v) = (u + u^2 + u^3) - v^3*(1 - 2*u + 4*u^2).

G.f. A(x) satisfies 0 = f(A(x), A(x^5)) where f(u, v) = u^6 + v^6 - 16*u^5*v^5 + 20*u^4*v^4 + 10*u^2*v^2*(u^3 + v^3) - 20*u^3*v^3 - 5*u*v*(u^3 + v^3) + 5*u^2*v^2 - u*v.

Expansion of b(q^2) / b(q) in powers of q where b() is a cubic AGM theta function.

G.f. is a period 1 Fourier series which satisfies f(-1 / (18 t)) = (1/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A062242.

a(n) = 3*A128129(n) unless n=0.

Convolution inverse of A141094. - Michael Somos, Feb 19 2015

a(n) ~ exp(2*sqrt(2*n)*Pi/3) / (2^(7/4) * sqrt(3) * n^(3/4)). - Vaclav Kotesovec, Oct 13 2015

EXAMPLE

G.f. = 1 + 3*q + 6*q^2 + 12*q^3 + 21*q^4 + 36*q^5 + 60*q^6 + 96*q^7 + ...

MATHEMATICA

a[ n_] := SeriesCoefficient[ QPochhammer[ q^2]^3 QPochhammer[ q^3] / (QPochhammer[ q]^3 QPochhammer[ q^6]), {q, 0, n}]; (* Michael Somos, Feb 19 2015 *)

nmax=60; CoefficientList[Series[Product[(1-x^(2*k))^3 * (1-x^(3*k)) / ((1-x^k)^3 * (1-x^(6*k))), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Oct 13 2015 *)

PROG

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

CROSSREFS

Cf. A062242, A128129, A141094.

Sequence in context: A115855 A234248 A294387 * A162920 A247662 A215005

Adjacent sequences:  A128125 A128126 A128127 * A128129 A128130 A128131

KEYWORD

nonn

AUTHOR

Michael Somos, Feb 15 2007

STATUS

approved

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Last modified December 12 15:11 EST 2019. Contains 329960 sequences. (Running on oeis4.)