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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. 3
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 (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 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

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: A147285 A147250 A066175 * A117079 A026745 A139333

Adjacent sequences:  A132975 A132976 A132977 * A132979 A132980 A132981

KEYWORD

nonn

AUTHOR

Michael Somos, Sep 07 2007

STATUS

approved

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Last modified May 20 20:16 EDT 2019. Contains 323426 sequences. (Running on oeis4.)