A227229 Expansion of (psi(q)^3 / psi(q^3))^2 in powers of q where psi() is a Ramanujan theta function.

1, 6, 15, 24, 33, 36, 33, 48, 69, 78, 90, 72, 51, 84, 120, 144, 141, 108, 87, 120, 198, 192, 180, 144, 87, 186, 210, 240, 264, 180, 198, 192, 285, 288, 270, 288, 105, 228, 300, 336, 414, 252, 264, 264, 396, 468, 360, 288, 159, 342, 465, 432, 462, 324, 249

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 8 and 32 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

Michael 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) + a(q^2))^2 / 4 in powers of q where a() is a cubic AGM theta function.

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

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

G.f. is a period 1 Fourier series which satisfies f(-1 / (6 t)) = (27/4) (t/i)^2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A227226.

Convolution square of A107760.

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

Example

G.f. = 1 + 6*q + 15*q^2 + 24*q^3 + 33*q^4 + 36*q^5 + 33*q^6 + 48*q^7 + 69*q^8 + ...

Mathematica

a[ n_] := SeriesCoefficient[ (QPochhammer[ q^3] QPochhammer[ q^2]^6 / (QPochhammer[q]^3 QPochhammer[q^6]^2))^2, {q, 0, n}];
a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, 0, q]^6 / EllipticTheta[ 2, 0, q^3]^2/16, {q, 0, 2 n}];
a[ n_] := If[ n < 1, Boole[ n == 0], 3 Sum[ {2, 3/2, 2, 3/2, 2, 0}[[ Mod[d, 6, 1]]] d, {d, Divisors[n]}]];
a[ n_] := If[ n < 1, Boole[ n == 0], 3 Sum[ {2, 1, 2, 1, 2, -8}[[ 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)^6 * eta(x^3 + A) / (eta(x + A)^3 * eta(x^6 + A)^2))^2, n))};
(Sage) A = ModularForms( Gamma0(6), 2, prec=50) . basis(); A[0] + 6*A[1] + 15*A[2];
(Magma) A := Basis( ModularForms( Gamma0(6), 2), 50); A[1] + 6*A[2] + 15*A[3];

Crossrefs

Cf. A107760, A227226.

Sequence in context: A238905 A187918 A190747 * A274319 A043477 A055040

Adjacent sequences: A227226 A227227 A227228 * A227230 A227231 A227232

Keyword

nonn

Author

Michael Somos, Sep 19 2013

Status

approved