A328797 Expansion of (chi(-x) * chi(x^3))^2 in powers of x where chi() is a Ramanujan theta function.

1, -2, 1, 0, 0, -2, 2, 0, 2, -2, 1, 0, 2, -6, 2, 0, 3, -6, 4, 0, 4, -8, 4, 0, 7, -14, 7, 0, 6, -16, 10, 0, 11, -20, 11, 0, 14, -32, 16, 0, 17, -38, 21, 0, 22, -46, 24, 0, 32, -66, 34, 0, 34, -78, 44, 0, 49, -96, 50, 0, 60, -130, 66, 0, 72, -154, 84, 0, 90, -186

Offset

0,2

Comments

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

Convolution square of A328800.

G.f. is a period 1 Fourier series which satisfies f(-1 / (36 t)) = 2 g(t) where q = exp(2 Pi i t) and g() is g.f. for A328790.

Links

Table of n, a(n) for n=0..69.

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

Expansion of q^(1/3) * (eta(q) * 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 [-2, 0, 0, 0, -2, -2, -2, 0, 0, 0, -2, 0, ...].

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

a(n) = (-1)^n * A328795(n). a(2*n) = A112206(n).

a(4*n) = A328789(n). a(4*n + 1) = -2 * A328798(n). a(4*n + 2) = A328790(n). a(4*n + 3) = 0.

Example

G.f. = 1 - 2*x + x^2 - 2*x^5 + 2*x^6 + 2*x^8 - 2*x^9 + x^10 + ...
G.f. = q^-1 - 2*q^2 + q^5 - 2*q^14 + 2*q^17 + 2*q^23 - 2*q^26 + ..

Mathematica

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

Prog

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

Crossrefs

Cf. A112206, A328789, A328790, A328795, A328798, A328800.

Sequence in context: A058626 A258278 A122856 * A328795 A353846 A055791

Adjacent sequences: A328794 A328795 A328796 * A328798 A328799 A328800

Keyword

sign

Author

Michael Somos, Oct 27 2019

Status

approved