A132978 Expansion of q^(-2/3) * (psi(-q^3) / psi(-q)^3) * (c(q^2) / 3) in powers of q where psi() is a Ramanujan theta function and c() is a cubic AGM theta function.

1, 3, 7, 15, 32, 63, 114, 201, 350, 591, 967, 1554, 2468, 3855, 5916, 8970, 13471, 20007, 29384, 42771, 61784, 88530, 125838, 177642, 249230, 347484, 481506, 663549, 909788, 1241127, 1684824, 2276781, 3063657, 4105275, 5478698, 7283709, 9648360, 12735471

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 q^(-2/3) * (psi(-q^3) / psi(-q)^3) * (c(q^2) / 3) in powers of q where psi() is a Ramanujan theta function and c() is a cubic AGM theta function.

Expansion of psi(-x^3)^3 * f(-x, x^2) / psi(-x)^4 in powers of x where psi(), f(,) are Ramanujan theta functions.

Expansion of q^(-2/3) * (eta(q^2) * eta(q^6))^2 * eta(q^3) * eta(q^12) / ( eta(q)* eta(q^4) )^3 in powers of q.

Euler transform of period 12 sequence [ 3, 1, 2, 4, 3, -2, 3, 4, 2, 1, 3, 0, ...].

a(n) = A132975(3*n + 2).

Convolution of A132974 and A045833.

a(n) ~ exp(2*Pi*sqrt(n/3)) / (2 * 3^(9/4) * n^(3/4)). - Vaclav Kotesovec, Oct 13 2015

Empirical: Sum_{n>=0} a(n) / exp(n*Pi) = (2/9) * exp(2*Pi/3) * Pi * Gamma(11/12) * (3^(1/2)-1)^2 / Gamma(2/3)^2 / Gamma(7/12) = A388606. - Simon Plouffe, Sep 18 2025

Example

G.f. = 1 + 3*x + 7*x^2 + 15*x^3 + 32*x^4 + 63*x^5 + 114*x^6 + 201*x^7 + ...
G.f. = q^2 + 3*q^5 + 7*q^8 + 15*q^11 + 32*q^14 + 63*q^17 + 114*q^20 + ...

Mathematica

a[ n_] := SeriesCoefficient[ 2^(1/2) x^(-5/8) EllipticTheta[ 3, 0, x^3]  QPochhammer[ x, -x] EllipticTheta[ 2, Pi/4, x^(3/2)]^3 / EllipticTheta[ 2, Pi/4, x^(1/2)]^4, {x, 0, n}] // Simplify;
nmax=60; CoefficientList[Series[Product[(1+x^(3*k))^3 * (1-x^(3*k))^4 * (1+x^(6*k)) / ( (1-x^k)^4 * (1+x^k) * (1+x^(2*k))^3), {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)^2 * eta(x^3 + A) * eta(x^6 + A)^2 * eta(x^12 + A) / ( eta(x + A) * eta(x^4 + A))^3, n))};

Crossrefs

Cf. A045833, A132974, A132975.

Sequence in context: A147250 A336701 A066175 * A336976 A368346 A117079

Adjacent sequences: A132975 A132976 A132977 * A132979 A132980 A132981

Keyword

nonn

Author

Michael Somos, Sep 07 2007

Status

approved