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

Author

Michael Somos, Sep 07 2007

Keywords

• nonn

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).

Examples

			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 + ...
		

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

Crossrefs

Cf. A045833, A132974, A132975.

Programs

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 *)

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))};

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