A229180 Expansion of (chi(-x) * chi(-x^3))^-3 in powers of x where chi() is a Ramanujan theta function.

1, 3, 6, 16, 33, 60, 118, 210, 354, 612, 1008, 1608, 2583, 4035, 6174, 9448, 14196, 21024, 31054, 45282, 65322, 93884, 133638, 188640, 265225, 370086, 512934, 708136, 971628, 1325724, 1802134, 2437200, 3280452, 4400132, 5876184, 7815288, 10360890, 13683525

Offset

0,2

Comments

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

In Verrill (1999) section 2.6, denoted by g as a function of q.

References

H. Verrill, Some Congruences related to modular forms, Max Planck Institute, 1999.

Links

G. C. Greubel, Table of n, a(n) for n = 0..2500

Michael Somos, Introduction to Ramanujan theta functions

H. Verrill, Some Congruences related to modular forms

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

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

Euler transform of period 6 sequence [3, 0, 6, 0, 3, 0, ...].

G.f. is a period 1 Fourier series which satisfies f(-1 / (24 t)) = (1/8) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A058492.

G.f.: t / (1 - 10*t^2 + 9*t^4)^(1/2) where t = the g.f. of A217786.

G.f.: 1 / (Product_{k>0} (1 - x^(2*k - 1)) * (1 - x^(6*k - 3)))^3.

Convolution inverse of A058492.

a(n) ~ exp(2*Pi*sqrt(n/3)) / (16 * 3^(1/4) * n^(3/4)). - Vaclav Kotesovec, Sep 07 2015

Example

G.f. = 1 + 3*x + 6*x^2 + 16*x^3 + 33*x^4 + 60*x^5 + 118*x^6 + 210*x^7 + ...
G.f. = q + 3*q^3 + 6*q^5 + 16*q^7 + 33*q^9 + 60*q^11 + 118*q^13 + 210*q^15 + ...

Mathematica

a[ n_] := SeriesCoefficient[ 1 / (QPochhammer[ x, x^2] QPochhammer[x^3, x^6])^3, {x, 0, n}];
nmax = 40; CoefficientList[Series[Product[1/((1 - x^(2*k - 1)) * (1 - x^(6*k - 3)))^3, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 07 2015 *)

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)))^3, n))};

Crossrefs

Cf. A058492, A084471, A217786.

Sequence in context: A291611 A308401 A196261 * A219810 A364319 A122742

Adjacent sequences: A229177 A229178 A229179 * A229181 A229182 A229183

Keyword

nonn

Author

Michael Somos, Sep 30 2013

Status

approved