A112280 - OEIS

%I #9 Sep 08 2022 08:45:21

%S 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,

%T 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,

%U 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

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

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

%H G. C. Greubel, <a href="/A112280/b112280.txt">Table of n, a(n) for n = 0..1000</a>

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

%e 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).

%e A(x)^(1/3) = 1 + 2*x - 4*x^2 + 15*x^3 - 60*x^4 + 268*x^5 -+...

%e 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)].

%p 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

%t 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 *)

%o (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)}

%o (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

%o (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

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

%K nonn

%O 0,2

%A _Paul D. Hanna_, Sep 01 2005