A328790 Expansion of (chi(x) / chi(-x^6))^2 in powers of x where chi() is a Ramanujan theta function.

1, 2, 1, 2, 4, 4, 7, 10, 11, 16, 21, 24, 34, 44, 50, 66, 84, 98, 125, 156, 181, 226, 277, 322, 397, 480, 557, 674, 807, 936, 1121, 1330, 1538, 1824, 2146, 2476, 2915, 3408, 3918, 4578, 5322, 6104, 7090, 8198, 9375, 10830, 12464, 14214, 16345, 18734, 21303

Offset

0,2

Comments

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

Convolution square of A328796.

G.f. is a period 1 Fourier series which satisfies f(-1 / (36 t)) = 1/2 g(t) where q = exp(2 Pi i t) and g() is g.f. for A328797.

Links

Table of n, a(n) for n=0..50.

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

Expansion of q^(-5/12) * (eta(q^2)^2 * eta(q^12))^2 / (eta(q) * eta(q^4) * eta(q^6))^2 in power of q.

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

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

a(n) = A112206(2*n + 1).

a(n) ~ exp(2*Pi*sqrt(n)/3) / (4*sqrt(3)*n^(3/4)). - Vaclav Kotesovec, Oct 31 2019

Example

G.f. = 1 + 2*x + x^2 + 2*x^3 + 4*x^4 + 4*x^5 + 7*x^6 + 10*x^7 + ...
G.f. = q^5 + 2*q^17 + q^29 + 2*q^41 + 4*q^53 + 4*q^65 + 7*q^77 + ...

Mathematica

a[ n_] := SeriesCoefficient[ (QPochhammer[ -x, x^2] QPochhammer[ -x^6, 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^12 + A))^2 / (eta(x + A) * eta(x^4 + A) * eta(x^6 + A))^2, n))};

Crossrefs

Cf. A112206, A328796, A328797.

Sequence in context: A132320 A353400 A076369 * A319773 A256188 A072727

Adjacent sequences: A328787 A328788 A328789 * A328791 A328792 A328793

Keyword

nonn

Author

Michael Somos, Oct 27 2019

Status

approved