A132107 Expansion of (f(x) / f(x^3))^6 in powers of x where f() is a Ramanujan theta function.

1, 6, 9, -16, -66, -54, 98, 300, 243, -364, -1128, -828, 1221, 3498, 2511, -3528, -9876, -6804, 9358, 25428, 17217, -23068, -61644, -40824, 53916, 141318, 92340, -119912, -310554, -199980, 256792, 656436, 418311, -530960, -1344144, -847584, 1066157, 2673372

Offset

0,2

Comments

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

Euler transform of period 12 sequence [ 6, -12, 0, -6, 6, 0, 6, -6, 0, -12, 6, 0, ...].

G.f. is a period 1 Fourier series which satisfies f(-1 / (48 t)) = 27 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A264026.

G.f.: (Product_{k>0} (1 + x^k) * (1 - x^(2*k)) * (1 + x^(3*k)) * (1 - x^(6*k)) / ((1 + x^(2*k)) * (1 + x^(6*k))))^6.

Example

G.f. = 1 + 6*x + 9*x^2 - 16*x^3 - 66*x^4 - 54*x^5 + 98*x^6 + 300*x^7 + 243*x^8 + ...
G.f. = 1/q + 6*q + 9*q^3 - 16*q^5 - 66*q^7 - 54*q^9 + 98*q^11 + 300*q^13 + ...

Mathematica

a[ n_] := SeriesCoefficient[ (QPochhammer[ -x] / QPochhammer[ -x^3])^6, {x, 0, n}]; (* Michael Somos, Aug 26 2015 *)

Prog

(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A)^3 * eta(x^3 + A) * eta(x^12 + A) / (eta(x + A) * eta(x^4 + A) * eta(x^6 + A)^3))^6, n))};

Crossrefs

Cf. A007262, A264026.

Sequence in context: A282498 A031326 A290791 * A007262 A129317 A316067

Adjacent sequences: A132104 A132105 A132106 * A132108 A132109 A132110

Keyword

sign

Author

Michael Somos, Aug 09 2007

Status

approved