A132301 Expansion of f(-x, -x^5) * f(-x)^2 / f(-x^6)^3 in powers of x where f(, ) and f() are Ramanujan theta functions.

1, -3, 1, 3, -1, 0, 1, -6, 0, 6, -3, 3, 4, -12, 1, 12, -6, 3, 5, -24, 1, 24, -10, 6, 11, -42, 4, 42, -19, 12, 17, -72, 4, 69, -31, 18, 31, -120, 9, 114, -50, 30, 46, -189, 11, 180, -79, 48, 77, -294, 21, 276, -122, 72, 112, -450, 28, 420, -183, 108, 173, -672

Offset

0,2

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..1000

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

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

Euler transform of period 6 sequence [ -3, -2, -2, -2, -3, 0, ...].

Given g.f. A(x), then B(q) = A(q^3) / (3*q) satisfies 0 = f(B(q), B(q^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 / (36 t)) = 6 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A132302.

a(2*n) = A132179(n). a(2*n + 1) = -3 * A092848(n). - Michael Somos, Nov 01 2015

Example

G.f. = 1 - 3*x + x^2 + 3*x^3 - x^4 + x^6 - 6*x^7 + 6*x^9 - 3*x^10 + 3*x^11 + ...
G.f. = 1/q - 3*q^2 + q^5 + 3*q^8 - q^11 + q^17 - 6*q^20 + 6*q^26 - 3*q^29 + ...

Mathematica

a[ n_] := SeriesCoefficient[ QPochhammer[ x, x^6] QPochhammer[ x^5, x^6] QPochhammer[ x]^2 / QPochhammer[ x^6]^2, {x, 0, n}]; (* Michael Somos, Nov 01 2015 *)

Prog

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

Crossrefs

Cf. A092848, A132179, A132302.

Sequence in context: A243332 A156872 A263526 * A073272 A268442 A175623

Adjacent sequences: A132298 A132299 A132300 * A132302 A132303 A132304

Keyword

sign

Author

Michael Somos, Aug 17 2007

Status

approved