A328800 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

Offset: 0

Author

Michael Somos, Oct 27 2019

Keywords

• sign

Comments

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

Convolution square is A328797.

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 A328796.

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, A328796, A328797, A328802.

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 + A) * eta(x^6 + A)^2) / (eta(x^2 + A) * eta(x^3 + A) * eta(x^12 + A)), n))};

Formula

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

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

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

a(n) = (-1)^n * A328802. a(4*n) = A097242(n). a(4*n + 1) = -A328796(n). a(4*n + 2) = a(4*n + 3) = 0.