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A181982 Expansion of (1 - x^2)^2 * (1 + x)^2 / (1 - x^6) in powers of x. 2
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

LINKS

G. C. Greubel, Table of n, a(n) for n = 0..5000

M. 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 [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.

EXAMPLE

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 + ...

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 *)

PROG

(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))))};

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

CROSSREFS

Sequence in context: A059147 A091891 A258127 * A070194 A323300 A105584

Adjacent sequences:  A181979 A181980 A181981 * A181983 A181984 A181985

KEYWORD

sign

AUTHOR

Michael Somos, Apr 04 2012

STATUS

approved

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Last modified January 24 04:35 EST 2020. Contains 331183 sequences. (Running on oeis4.)