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

0,25

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.

Links

Table of n, a(n) for n=0..84.

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

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.

Example

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 + ...

Mathematica

a[ n_] := SeriesCoefficient[ QPochhammer[ -x, x^2] QPochhammer[ x^3, x^6], {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)) / (eta(x + A) * eta(x^4 + A) * eta(x^6 + A)), n))};

Crossrefs

Cf. A097242, A328795, A328796, A328800.

Sequence in context: A263834 A263754 A328800 * A355245 A103822 A225927

Adjacent sequences: A328799 A328800 A328801 * A328803 A328804 A328805

Keyword

nonn

Author

Michael Somos, Oct 28 2019

Status

approved