A328802 Expansion of chi(x) * chi(-x^3) in powers of x where chi() is a Ramanujan theta function.

1, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 2, 2, 0, 0, 1, 2, 0, 0, 2, 2, 0, 0, 3, 3, 0, 0, 3, 3, 0, 0, 3, 3, 0, 0, 5, 5, 0, 0, 4, 5, 0, 0, 6, 5, 0, 0, 7, 7, 0, 0, 7, 8, 0, 0, 8, 8, 0, 0, 11, 11, 0, 0, 10, 12, 0, 0, 13, 12, 0, 0, 15

Offset: 0

Author

Michael Somos, Oct 28 2019

Keywords

• nonn

Comments

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

Convolution square is A328795.

G.f. is a period 1 Fourier series which satisfies f(-1 / (1728 t)) = 2^(1/2) g(t) where q = exp(2 Pi i t) and g() is g.f. for A097242.

Examples

			G.f. = 1 + x + x^5 + x^8 + x^12 + x^13 + x^16 + x^17 + x^20 + ...
G.f. = q^-1 + q^5 + q^29 + q^47 + q^71 + q^77 + q^95 + q^101 + ...
		

Links

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

Crossrefs

Cf. A097242, A328795, A328796, A328800.

Programs

Mathematica

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

PARI

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

Formula

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

Euler transform of period 12 sequence [1, -1, 0, 0, 1, -1, 1, 0, 0, -1, 1, 0, ...].

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

a(n) = (-1)^n * A328800. a(4*n) = A097242(n). a(4*n + 1) = A328796(n). a(4*n + 2) = a(4*n + 3) = 0.