A323202 Expansion of (1 - x) * (1 - x^3) / (1 - x^4) in powers of x.

1, -1, 0, -1, 2, -1, 0, -1, 2, -1, 0, -1, 2, -1, 0, -1, 2, -1, 0, -1, 2, -1, 0, -1, 2, -1, 0, -1, 2, -1, 0, -1, 2, -1, 0, -1, 2, -1, 0, -1, 2, -1, 0, -1, 2, -1, 0, -1, 2, -1, 0, -1, 2, -1, 0, -1, 2, -1, 0, -1, 2, -1, 0, -1, 2, -1, 0, -1, 2, -1, 0, -1, 2, -1, 0

Offset: 0

Author

Michael Somos, Jan 06 2019

Keywords

• sign
• easy

Examples

			G.f. = 1 - x - x^3 + 2*x^4 - x^5 - x^7 + 2*x^8 - x^9 - x^11 + ...
		

Links

• Michael Somos, Rational Function Multiplicative Coefficients.
• Index entries for linear recurrences with constant coefficients, signature (-1,-1,-1).

Crossrefs

Cf. A098178.

Programs

Mathematica

a[ n_] := (-1)^n + If[Mod[n, 2] == 0, (-1)^(n/2), 0] - Boole[n == 0];
a[ n_] := {-1, 0, -1, 2}[[Mod[n, 4, 1]]] - Boole[n == 0];
a[ n_] := SeriesCoefficient[ (1 - x) (1 - x^3) / (1 - x^4), {x, 0, Abs@n}];
LinearRecurrence[{-1,-1,-1},{1,-1,0,-1},80] (* Harvey P. Dale, May 31 2021 *)

PARI

{a(n) = (-1)^n + if(n%2==0, (-1)^(n/2)) - (n==0)};

PARI

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

PARI

{a(n) = n = abs(n); polcoeff( (1 - x) * (1 - x^3) / (1 - x^4) + x * O(x^n), n)};

PARI

{a(n) = my(e); n=abs(n); if( n<1, n==0, e=valuation(n, 2); -if( e<2, 1-e, -2))};

Formula

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

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

Moebius transform is length 4 sequence [-1, 1, 0, 2].

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

a(n) = a(-n) for all n in Z. a(n+2) = a(n-2) except if n=2 or n=-2.

a(n) = (-1)^n * A098178(n), a(2*n + 1) = -1, a(4*n + 2) = 0 for all n in Z.

E.g.f.: cos(x) + exp(-x) - 1. - Stefano Spezia, Aug 04 2025