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A164357
Expansion of (1 - x^2)^5 / ((1 - x)^4 * (1 - x^6)) in powers of x.
1
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
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
KEYWORD
sign,easy
AUTHOR
Michael Somos, Aug 13 2009
STATUS
approved