A181982 Expansion of (1 - x^2)^2 * (1 + x)^2 / (1 - x^6) in powers of x.

1, 2, -1, -4, -1, 2, 2, 2, -1, -4, -1, 2, 2, 2, -1, -4, -1, 2, 2, 2, -1, -4, -1, 2, 2, 2, -1, -4, -1, 2, 2, 2, -1, -4, -1, 2, 2, 2, -1, -4, -1, 2, 2, 2, -1, -4, -1, 2, 2, 2, -1, -4, -1, 2, 2, 2, -1, -4, -1, 2, 2, 2, -1, -4, -1, 2, 2, 2, -1, -4, -1, 2, 2, 2

Offset: 0

Author

Michael Somos, Apr 04 2012

Keywords

• sign

Examples

			G.f. = 1 + 2*x - x^2 - 4*x^3 - x^4 + 2*x^5 + 2*x^6 + 2*x^7 - x^8 - 4*x^9 + ...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..5000
• Michael Somos, Rational Function Multiplicative Coefficients
• Index entries for linear recurrences with constant coefficients, signature (0, -1, 0, -1).

Programs

Magma

m:=50; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((1-x^2)^2*(1+x)^2/(1-x^6))); // G. C. Greubel, Aug 12 2018

Mathematica

CoefficientList[Series[(1-x^2)^2*(1+x)^2/(1-x^6), {x, 0, 50}], x] (* G. C. Greubel, Aug 12 2018 *)
a[ n_] := {2, -1, -4, -1, 2, 2}[[Mod[n, 6, 1]]] - Boole[n==0]; (* Michael Somos, Nov 14 2019 *)

PARI

{a(n) = [2, 2, -1, -4, -1, 2][n%6 + 1] - (n == 0)};

PARI

{a(n) = if( n<0, a(-n), polcoeff( (1 + x)^2 * (1 - x^2)^2 / (1 - x^6) + x * O(x^n), n))};

PARI

{a(n) = if( n<0, a(-n), if( n<1, n==0, sumdiv( n, d, 2*(d==1) - 3*(d==2) - 6*(d==3) + 9*(d==6))))};

Formula

Euler transform of length 6 sequence [2, -4, 0, 0, 0, 1].

a(n) = 2 * b(n) where b(n) is multiplicative with b(2^e) = -1/2 if e>0, b(3^e) = -2 if e>0, b(p^e) = 1 if p>3.

Expansion of 1 + 2*x / (1 - x) - 3*x^2 / (1 - x^2) - 6*x^3 / (1 - x^3) + 9*x^6 / (1 - x^6) in powers of x.

G.f.: (1 - x) * (1 + x)^3 / (1 + x^2 + x^4) = (1 - x^2)^4 / ((1 - x)^2 * (1 - x^6)).

a(-n) = a(n). a(n - 6) = a(n) unless n=0. a(n) = -a(n-2) - a(n-4) unless n=0, 2, 4.