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

Author

Michael Somos, Sep 30 2013

Keywords

• nonn

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.

Examples

			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 + ...
		

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

Crossrefs

Cf. A058492, A084471, A217786.

Programs

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 *)

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))};

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