A233006 Expansion of psi(x) / f(-x^6) in powers of x where psi(), f() are Ramanujan theta functions.

1, 1, 0, 1, 0, 0, 2, 1, 0, 1, 1, 0, 3, 2, 0, 3, 1, 0, 5, 3, 0, 5, 2, 0, 8, 5, 0, 8, 4, 0, 12, 7, 0, 12, 6, 0, 19, 11, 0, 19, 9, 0, 27, 15, 0, 28, 14, 0, 39, 22, 0, 41, 20, 0, 55, 31, 0, 58, 29, 0, 77, 43, 0, 82, 41, 0, 106, 58, 0, 113, 57, 0, 145, 80, 0, 156

Offset: 0

Author

Michael Somos, Dec 03 2013

Keywords

• nonn

Comments

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

Examples

			G.f. = 1 + x + x^3 + 2*x^6 + x^7 + x^9 + x^10 + 3*x^12 + 2*x^13 + 3*x^15 + ...
G.f. = q + q^9 + q^25 + 2*q^49 + q^57 + q^73 + q^81 + 3*q^97 + 2*q^105 + ...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..2500
• Michael Somos, Introduction to Ramanujan theta functions
• Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Crossrefs

Cf. A070047, A097451.

Programs

Mathematica

a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, 0, x^(1/2)] / (2 x^(1/8) QPochhammer[ x^6]), {x, 0, n}];

PARI

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

Formula

Expansion of q^(-1/8) * eta(q^2)^2 / (eta(q) * eta(q^6)) in powers of q.

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

G.f. is a period 1 Fourier series which satisfies f(-1 / (1152 t)) = (3/2)^(1/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. of A070047.

G.f.: Product_{k>0} (1 + x^k) / (1 + x^(2*k) + x^(4*k)).

a(3*n) = A070047(n). a(3*n + 1) = A097451(n). a(3*n + 2) = 0.