A137566 Expansion of f(-x, -x^5) / f(-x^6)^2 in powers of x where f(, ) is Ramanujan's general theta function.

1, -1, 0, 0, 0, -1, 2, -2, 1, 0, 0, -2, 5, -5, 2, 0, 1, -5, 10, -10, 5, -1, 2, -10, 20, -20, 10, -2, 5, -20, 36, -36, 20, -6, 10, -36, 65, -65, 36, -12, 21, -65, 110, -110, 65, -25, 38, -110, 185, -185, 110, -46, 70, -185, 300, -300, 186, -85, 120, -300, 481

Offset

0,7

Links

Table of n, a(n) for n=0..60.

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

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

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

G.f. is a period 1 Fourier series which satisfies f(-1 / (864 t)) = 24^(-1/2) (t/i)^(-1/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A007690.

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

Example

G.f. = 1 - x - x^5 + 2*x^6 - 2*x^7 + x^8 - 2*x^11 + 5*x^12 - 5*x^13 + 2*x^14 + ...
G.f. = 1/q - q^5 - q^29 + 2*q^35 - 2*q^41 + q^47 - 2*q^65 + 5*q^71 - 5*q^77 + ...

Mathematica

a[ n_] := SeriesCoefficient[ QPochhammer[ x] / (QPochhammer[ x^2] QPochhammer[ x^3]), {x, 0, n}]; (* Michael Somos, Oct 12 2015 *)

Prog

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

Crossrefs

Cf. A007690.

Sequence in context: A181169 A029390 A108040 * A258277 A122865 A246656

Adjacent sequences: A137563 A137564 A137565 * A137567 A137568 A137569

Keyword

sign

Author

Michael Somos, Jan 26 2008

Status

approved