A112280 Coefficients, read modulo 9, of the cube of q-series (q;q)_oo.

1, 6, 0, 5, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 7, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 8, 0, 0, 0, 0, 0, 0, 0, 0, 8, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 7, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset

0,2

Comments

The cube-root of g.f. A(x) is an integer series (A112281).

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

Formula

G.f.: A(x) = Sum_{n>=0} A112282(n) * x^(n*(n+1)/2) where A112282(n) = (-1)^n*(2*n+1) (mod 9).

Example

A(x) = 1 + 6*x + 5*x^3 + 2*x^6 + 0*x^10 + 7*x^15 + 4*x^21 +... = (1 - 3*x + 5*x^3 - 7*x^6 + 9*x^10 - 11*x^15 +...) (mod 9).
A(x)^(1/3) = 1 + 2*x - 4*x^2 + 15*x^3 - 60*x^4 + 268*x^5 -+...
Notation: q-series (q;q)_oo = Product_{n>=1} (1-q^n) = 1 + Sum_{n>=1} (-1)^n*[q^(n*(3*n-1)/2) + q^(n*(3*n+1)/2)].

Maple

seq(coeff(series( add(`mod`((-1)^n*(2*n+1), 9)*x^(n*(n+1)/2), n = 0 .. 140), x, n+1), x, n), n = 0 .. 120); # G. C. Greubel, Nov 05 2019

Mathematica

CoefficientList[Series[ Sum[Mod[(-1)^n*(2*n+1), 9]* x^(n(n+1)/2), {n, 0, 140}] , {x, 0, 120}], x] (* G. C. Greubel, Nov 05 2019 *)

Prog

(PARI) {a(n)=polcoeff(sum(k=0, sqrtint(2*n+1), (((-1)^k*(2*k+1))%9)*x^(k*(k+1)/2)+x*O(x^n)), n)}
(Magma) R<x>:=PowerSeriesRing(Integers(), 120); Coefficients(R!( (&+[((-1)^n*(2*n+1) mod 9)*x^Binomial(n+1, 2): n in [0..140]]) )); // G. C. Greubel, Nov 05 2019
(Sage) [ (sum(((-1)^n*(2*n+1)%9) *x^(n*(n+1)/2) for n in (0..140)) ).series(x, n+1).list()[n] for n in (0..120)]  # G. C. Greubel, Nov 05 2019

Crossrefs

Cf. A112281 (A(x)^(1/3)), A112282 (nonzero terms), A111983 (variant).

Sequence in context: A196623 A265275 A113024 * A204850 A202394 A202954

Adjacent sequences: A112277 A112278 A112279 * A112281 A112282 A112283

Keyword

nonn

Author

Paul D. Hanna, Sep 01 2005

Status

approved