A328798 Expansion of 1 / (chi(-x) * chi(-x^3)) in powers of x where chi() is a Ramanujan theta function.

1, 1, 1, 3, 3, 4, 7, 8, 10, 16, 19, 23, 33, 39, 48, 65, 77, 93, 122, 144, 173, 220, 259, 309, 384, 451, 534, 653, 764, 899, 1085, 1264, 1479, 1765, 2048, 2385, 2820, 3260, 3778, 4432, 5105, 5891, 6864, 7879, 9056, 10491, 12002, 13744, 15839, 18064, 20616, 23648

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 inverse is A112175, 2nd power is A102315, 3rd power is A229180, 6th power is A123653.

f(-1 / (216 t)) = 1/2 g(t) where q = exp(2 Pi i t) and g() is g.f. for A112175.

Examples

			G.f. = 1 + x + x^2 + 3*x^3 + 3*x^4 + 4*x^5 + 7*x^6 + 8*x^7 + ...
G.f. = q + q^7 + q^13 + 3*q^19 + 3*q^25 + 4*q^31 + 7*q^37 + ...
		

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..10000
• Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Crossrefs

Cf. A102315, A112175, A123653, A229180.

Programs

Mathematica

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

PARI

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

Formula

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

Euler transform of period 6 sequence [1, 0, 2, 0, 1, 0, ...].

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

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