A328789 Expansion of (chi(x^3) / chi(-x^2))^2 in powers of x where chi() is a Ramanujan theta function.

1, 0, 2, 2, 3, 4, 7, 6, 11, 14, 17, 22, 32, 34, 49, 60, 72, 90, 117, 132, 171, 206, 245, 298, 369, 422, 522, 620, 728, 868, 1043, 1198, 1439, 1688, 1962, 2304, 2717, 3114, 3668, 4258, 4909, 5698, 6627, 7566, 8788, 10112, 11574, 13310, 15317, 17410, 20010

Offset: 0

Author

Michael Somos, Oct 27 2019

Keywords

• nonn

Comments

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

Convolution square of A097242.

G.f. is a period 1 Fourier series which satisfies f(-1 / (36 t)) = 1/2 g(t) where q = exp(2 Pi i t) and g() is g.f. for A328795.

Examples

			G.f. = 1 + 2*x^2 + 2*x^3 + 3*x^4 + 4*x^5 + 7*x^6 + 6*x^7 + 11*x^8 + ...
G.f. = q^-1 + 2*q^23 + 2*q^35 + 3*q^47 + 4*q^59 + 7*q^71 + 6*q^83 + ...
		

Links

• Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Crossrefs

Cf. A097242, A112206, A328795.

Programs

Mathematica

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

PARI

{a(n) = my(A); if( n < 0, 0, A = x * O(x^n); polcoeff( (eta(x^4 + A) * eta(x^6 + A)^2)^2 / (eta(x^2 + A) * eta(x^3 + A) * eta(x^12 + A))^2, n))};

Formula

Expansion of q^(1/12) * (eta(q^4) * eta(q^6)^2)^2 / (eta(q^2) * eta(q^3) * eta(q^12))^2 in powers of q.

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

G.f.: Product_{k>=1} (1 + x^(6*k - 3))^2 / (1 - x^(4*k - 2))^2.

a(n) = A112206(2*n).

a(n) ~ exp(2*Pi*sqrt(n)/3) / (4*sqrt(3)*n^(3/4)). - Vaclav Kotesovec, Oct 31 2019