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

0,4

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.

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

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

Example

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 + ...

Mathematica

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

Prog

(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))};

Crossrefs

Cf. A102315, A112175, A123653, A229180.

Sequence in context: A152651 A343396 A091973 * A327726 A050066 A050064

Adjacent sequences: A328795 A328796 A328797 * A328799 A328800 A328801

Keyword

nonn

Author

Michael Somos, Oct 28 2019

Status

approved