A112281 The cube-root of the g.f. of A112280, which is congruent modulo 9 to the cube of q-series (q;q)_oo.

1, 2, -4, 15, -60, 268, -1275, 6322, -32280, 168525, -895272, 4823088, -26284036, 144623580, -802297080, 4482108215, -25193038332, 142365182220, -808318895340, 4608847319040, -26378042959008, 151485697164867, -872650786462376, 5041141102888080

Offset

0,2

Comments

G.f. A(x) at x=q is congruent modulo 3 to q-series (q;q)_oo.

Links

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

Formula

Limit a(n)/a(n+1) = z = -0.1630599902691518961128975774567541135944... where A(z) = 0.

Example

A(x) = 1 + 2*x - 4*x^2 + 15*x^3 - 60*x^4 + 268*x^5 -+...
= (1 - x - x^2 + x^5 + x^7 - x^12 - x^15 + x^22 +...) (mod 3).
A(x)^3 = 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).
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 .. 40))^(1/3), x, n+1), x, n), n = 0 .. 30); # G. C. Greubel, Nov 05 2019

Mathematica

CoefficientList[Series[(Sum[ Mod[(-1)^n*(2*n + 1), 9]*x^(n(n+1)/2), {n, 0, 50}])^(1/3), {x, 0, 30}], 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))^(1/3), n)}
(Sage) [( (sum(((-1)^n*(2*n+1)%9) *x^(n*(n+1)/2) for n in (0..40)))^(1/3) ).series(x, n+1).list()[n] for n in (0..30)] # G. C. Greubel, Nov 05 2019

Crossrefs

Cf. A112280 (A(x)^3).

Sequence in context: A188228 A243796 A153939 * A316661 A014517 A020110

Adjacent sequences: A112278 A112279 A112280 * A112282 A112283 A112284

Keyword

sign

Author

Paul D. Hanna, Sep 01 2005

Status

approved