A328795 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, 98, 0, 117, 244

Offset

0,2

Comments

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

Convolution square of A328802.

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 A328789.

Links

Table of n, a(n) for n=0..73.

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

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

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

a(n) = (-1)^n * A328797(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^2 + A)^2 * eta(x^3 + A))^2 / (eta(x+ A) * eta(x^4 + A) * eta(x^6 + A))^2, n))};

Crossrefs

Cf. A112206, A328789, A328790, A328797, A328798, A328802.

Sequence in context: A258278 A122856 A328797 * A353846 A055791 A245842

Adjacent sequences: A328792 A328793 A328794 * A328796 A328797 A328798

Keyword

nonn

Author

Michael Somos, Oct 28 2019

Status

approved