A128128 Expansion of chi(-q^3) / chi^3(-q) in powers of q where chi() is a Ramanujan theta function.

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

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

Michael 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 A337462

Adjacent sequences: A128125 A128126 A128127 * A128129 A128130 A128131

Keyword

nonn

Author

Michael Somos, Feb 15 2007

Status

approved