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A132974 Expansion of psi(-q^3) / psi(-q)^3 in powers of q where psi() is a Ramanujan theta function. 8
1, 3, 6, 12, 24, 45, 78, 132, 222, 363, 576, 900, 1392, 2121, 3180, 4716, 6936, 10098, 14550, 20796, 29520, 41595, 58176, 80856, 111750, 153561, 209820, 285240, 385968, 519840, 696960, 930516, 1237470, 1639314, 2163456, 2845080, 3728904, 4871211 (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).

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

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

G.f.: Product_{k>0} (1 - x^(3*k)) * (1 + x^(6*k)) / ( (1 - x^k) * (1 + x^(2*k)) )^3.

G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = (108)^(-1/2) (t/i)^(-1) g(t) where q = exp(2 Pi i t) and g(t) is the g.f. for A133637.

A132979(n) = (-1)^n * a(n). Convolution inverse of A132973.

a(n) ~ exp(2*Pi*sqrt(n/3)) / (2 * 3^(5/4) * n^(5/4)). - Vaclav Kotesovec, Oct 13 2015

EXAMPLE

G.f. = 1 + 3*q + 6*q^2 + 12*q^3 + 24*q^4 + 45*q^5 + 78*q^6 + 132*q^7 + ...

MATHEMATICA

a[ n_] := SeriesCoefficient[ 2 EllipticTheta[ 2, Pi/4, q^(3/2)] / EllipticTheta[ 2, Pi/4, q^(1/2)]^3 , {q, 0, n}]; (* Michael Somos, Sep 26 2017 *)

nmax=60; CoefficientList[Series[Product[(1-x^(3*k)) * (1+x^(6*k)) / ( (1-x^k)^3 * (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)^3 * eta(x^3 + A) * eta(x^12 + A ) / (eta(x + A)^3 * eta(x^4 + A)^3 * eta(x^6 + A)), n))};

CROSSREFS

Cf. A132973, A132979.

Sequence in context: A079079 A283839 A336758 * A132979 A163314 A018183

Adjacent sequences:  A132971 A132972 A132973 * A132975 A132976 A132977

KEYWORD

nonn

AUTHOR

Michael Somos, Sep 07 2007

STATUS

approved

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Last modified January 25 00:44 EST 2022. Contains 350565 sequences. (Running on oeis4.)