A164357 Expansion of (1 - x^2)^5 / ((1 - x)^4 * (1 - x^6)) in powers of x.

1, 4, 5, 0, -5, -4, 0, 4, 5, 0, -5, -4, 0, 4, 5, 0, -5, -4, 0, 4, 5, 0, -5, -4, 0, 4, 5, 0, -5, -4, 0, 4, 5, 0, -5, -4, 0, 4, 5, 0, -5, -4, 0, 4, 5, 0, -5, -4, 0, 4, 5, 0, -5, -4, 0, 4, 5, 0, -5, -4, 0, 4, 5, 0, -5, -4, 0, 4, 5, 0, -5, -4, 0, 4, 5, 0, -5, -4

Offset

0,2

Links

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

Michael Somos, Rational Function Multiplicative Coefficients

Index entries for linear recurrences with constant coefficients, signature (0,-1,0,-1).

Formula

Euler transform of length 6 sequence [4, -5, 0, 0, 0, 1].

a(n) = 4*b(n) unless n=0 where b() is multiplicative with b(2^e) = -5/4 * (-1)^e if e>1, b(3^e) = 0^e, b(p^e) = 1 if p == 1 (mod 6), b(p^e) = (-1)^e if p == 5 (mod 6).

a(n) = -a(-n) unless n=0. a(3*n) = 0 unless n=0. a(n+6) = a(n) unless n=0 or n=-6.

G.f.: 1 + (x/2) * ( 9 / (1 - x + x^2) - 1 / (1 + x + x^2)).

0 = a(n-2) + a(n) + a(n+2) unless n = 0, -2, 2. - Michael Somos, Jan 07 2019

Example

G.f. = 1 + 4*x + 5*x^2 - 5*x^4 - 4*x^5 + 4*x^7 + 5*x^8 - 5*x^10 - 4*x^11 + ...

Mathematica

a[ n_] := Boole[n == 0] - 1/2 KroneckerSymbol[-3, n] + 9/2 KroneckerSymbol[9, n] (-1)^Quotient[n, 3]; (* Michael Somos, Apr 17 2015 *)
CoefficientList[Series[(1 - x^2)^5/((1 - x)^4*(1 - x^6)), {x, 0, 100}], x] (* Wesley Ivan Hurt, Jan 31 2017 *)
a[ n_] := (-1)^Boole[n < 0] SeriesCoefficient[ 1 + (x/2) (9 / (1 - x + x^2) - 1 / (1 + x + x^2)), {x, 0, Abs@n}]; (* Michael Somos, Jan 07 2019 *)

Prog

(PARI) {a(n) = (n==0) - 1/2 * kronecker(-3, n) + 9/2 * kronecker(9, n) * (-1)^(n\3)};

Crossrefs

Sequence in context: A187045 A186930 A159567 * A092487 A322505 A192041

Adjacent sequences: A164354 A164355 A164356 * A164358 A164359 A164360

Keyword

sign,easy

Author

Michael Somos, Aug 13 2009

Status

approved