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A227226 Expansion of phi(-q^3)^6 / phi(-q)^2 where phi() is a Ramanujan theta function. 3
1, 4, 12, 20, 28, 24, 28, 32, 60, 68, 72, 48, 44, 56, 96, 120, 124, 72, 76, 80, 168, 160, 144, 96, 76, 124, 168, 212, 224, 120, 168, 128, 252, 240, 216, 192, 92, 152, 240, 280, 360, 168, 224, 176, 336, 408, 288, 192, 140, 228, 372, 360, 392, 216, 220, 288 (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).

Number 13 and 34 of the 126 eta-quotients listed in Table 1 of Williams 2012. - Michael Somos, Nov 10 2018

LINKS

G. C. Greubel, Table of n, a(n) for n = 0..2500

M. Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

K. S. Williams, Fourier series of a class of eta quotients, Int. J. Number Theory 8 (2012), no. 4, 993-1004.

FORMULA

Expansion of (a(q) + 2*a(q^2))^2 / 9 in powers of q where a(q) is a cubic AGM theta function.

Expansion of c(q)^4 / (3 * c(q^2))^2 in powers of q where c(q) is a cubic AGM theta function.

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

Euler transform of period 6 sequence [4, 2, -8, 2, 4, -4, ...].

G.f. is a period 1 Fourier series which satisfies f(-1 / (6 t)) = (16/3) (t/i)^2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A227229.

Convolution square of A123330.

EXAMPLE

G.f. = 1 + 4*q + 12*q^2 + 20*q^3 + 28*q^4 + 24*q^5 + 28*q^6 + 32*q^7 + ...

MATHEMATICA

a[ n_] := SeriesCoefficient[ (QPochhammer[ q^2] QPochhammer[ q^3]^6 / (QPochhammer[ q]^2 QPochhammer[ q^6]^3))^2, {q, 0, n}];

a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, q^3]^6 / EllipticTheta[ 4, 0, q]^2, {q, 0, n}];

a[ n_] := If[ n < 1, Boole[ n == 0], 4 Sum[ {1, 1, 4/3, 1, 1, 0}[[ Mod[d, 6, 1]]] d, {d, Divisors[n]}]];

a[ n_] := If[ n < 1, Boole[ n == 0], 4 Sum[ {1, 1, 2, 1, 1, -6}[[ Mod[d, 6, 1]]] n/d, {d, Divisors[n]}]];

PROG

(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A) * eta(x^3 + A)^6 / (eta(x + A)^2 * eta(x^6 + A)^3))^2, n))};

(Sage) A = ModularForms( Gamma0(6), 2, prec=50) . basis(); A[0] + 4*A[1] + 12*A[2];

(MAGMA) A := Basis( ModularForms( Gamma0(6), 2), 50); A[1] + 4*A[2] + 12*A[3];

CROSSREFS

Cf. A123330, A227229.

Sequence in context: A100717 A285526 A321466 * A242118 A030387 A269931

Adjacent sequences:  A227223 A227224 A227225 * A227227 A227228 A227229

KEYWORD

nonn

AUTHOR

Michael Somos, Sep 19 2013

STATUS

approved

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Last modified July 4 16:24 EDT 2020. Contains 335448 sequences. (Running on oeis4.)