A164356 Expansion of (1 - x^2)^4 / ((1 - x)^4 * (1 - x^4)) in powers of x.

1, 4, 6, 4, 2, 4, 6, 4, 2, 4, 6, 4, 2, 4, 6, 4, 2, 4, 6, 4, 2, 4, 6, 4, 2, 4, 6, 4, 2, 4, 6, 4, 2, 4, 6, 4, 2, 4, 6, 4, 2, 4, 6, 4, 2, 4, 6, 4, 2, 4, 6, 4, 2, 4, 6, 4, 2, 4, 6, 4, 2, 4, 6, 4, 2, 4, 6, 4, 2, 4, 6, 4, 2, 4, 6, 4, 2, 4, 6, 4, 2, 4, 6, 4, 2, 4, 6, 4, 2, 4, 6, 4, 2, 4, 6, 4, 2, 4, 6, 4, 2, 4, 6, 4, 2

Offset: 0

Author

Michael Somos, Aug 13 2009

Keywords

• nonn
• easy

Examples

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

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 (1,-1,1).

Crossrefs

Cf. A068073.

Programs

Mathematica

a[ n_] := -Boole[n == 0] + 4 - If[ EvenQ[n], (-1)^(n/2) 2, 0]; (* Michael Somos, Apr 17 2015 *)
a[ n_] := SeriesCoefficient[ -1 + 4/(1 - x) - 2/(1 + x^2), {x, 0, Abs@n}]; (* Michael Somos, Jan 07 2019 *)
LinearRecurrence[{1,-1,1},{1,4,6,4},120] (* or *) PadRight[{1},120,{2,4,6,4}] (* Harvey P. Dale, Aug 30 2024 *)

PARI

{a(n) = -(n==0) + 4 - if( n%2 == 0, (-1)^(n/2) * 2, 0)};

Formula

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

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

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

a(n) = a(-n) for all n in Z. a(n+4) = a(n) unless n=0 or n=-4. a(2*n + 1) = 4. a(4*n) = 2 unless n=0. a(4*n + 2) = 6.

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

a(n) = 2 * A068073(n) unless n=0. - Michael Somos, Apr 17 2015