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

0,7

Comments

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

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

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.

Example

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 + ...

Mathematica

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

Prog

(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))};

Crossrefs

Cf. A070047, A097451.

Sequence in context: A255361 A341685 A069844 * A145152 A124327 A346837

Adjacent sequences: A233003 A233004 A233005 * A233007 A233008 A233009

Keyword

nonn

Author

Michael Somos, Dec 03 2013

Status

approved