A132180 Expansion of f(q, q^2) * f(-q^3) / f(-q^2)^2 in powers of q where f(, ), f() are Ramanujan theta functions.

1, 1, 3, 1, 6, 3, 12, 5, 21, 10, 36, 15, 60, 26, 96, 39, 150, 63, 228, 92, 342, 140, 504, 201, 732, 295, 1050, 415, 1488, 591, 2088, 818, 2901, 1140, 3996, 1554, 5460, 2126, 7404, 2861, 9972, 3855, 13344, 5126, 17748, 6816, 23472, 8970, 30876, 11793, 40413

Offset: 0

Author

Michael Somos, Aug 12 2007

Keywords

• nonn

Comments

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

Examples

			G.f. = 1 + q + 3*q^2 + q^3 + 6*q^4 + 3*q^5 + 12*q^6 + 5*q^7 + 21*q^8 + 10*q^9 + ...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Michael Somos, Introduction to Ramanujan theta functions
• Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Crossrefs

Cf. A128128, A132179, A132302.

Programs

Mathematica

a[ n_] := SeriesCoefficient[ QPochhammer[ q^3]^3 / (QPochhammer[ q] QPochhammer[ q^2] QPochhammer[ q^6]), {q, 0, n}]; (* Michael Somos, Apr 26 2015 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ -q, q^3] QPochhammer[ -q^2, q^3] QPochhammer[ q^3]^2 / QPochhammer[ q^2]^2, {q, 0, n}]; (* Michael Somos, Nov 01 2015 *)

PARI

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

Formula

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

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

G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = (v^2 - 2*u)^3 - u^4 * (2*u - 3*v^2) * (4*u - 3*v^2).

G.f. is a period 1 Fourier series which satisfies f(-1 / (6 t)) = (2/3) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A132179.

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

a(2*n) = A128128(n). a(2*n + 1) = A132302(n).