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A163812 Expansion of (1 - x^5) * (1 - x^6) / ((1 - x) * (1 - x^10)) in powers of x. 4

%I #16 Dec 06 2019 12:27:06

%S 1,1,1,1,1,0,-1,-1,-1,-1,0,1,1,1,1,0,-1,-1,-1,-1,0,1,1,1,1,0,-1,-1,-1,

%T -1,0,1,1,1,1,0,-1,-1,-1,-1,0,1,1,1,1,0,-1,-1,-1,-1,0,1,1,1,1,0,-1,-1,

%U -1,-1,0,1,1,1,1,0,-1,-1,-1,-1,0,1,1,1,1,0,-1,-1,-1,-1,0,1,1,1,1,0,-1,-1,-1,-1,0,1,1,1,1,0,-1,-1,-1,-1,0,1,1,1,1

%N Expansion of (1 - x^5) * (1 - x^6) / ((1 - x) * (1 - x^10)) in powers of x.

%H G. C. Greubel, <a href="/A163812/b163812.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (1,-1,1,-1).

%F Euler transform of length 10 sequence [ 1, 0, 0, 0, -1, -1, 0, 0, 0, 1].

%F a(5*n) = 0 unless n=0.

%F a(n) = -a(-n) unless n=0. a(n+5) = -a(n) unless n=0 or n=-5.

%F G.f.: (1 + x^2 + x^4) / (1 - x + x^2 - x^3 + x^4).

%F a(n) = (-1)^n * A163818(n). Convolution inverse of A163811.

%F G.f.: A(x) = 1 / (1 - x / ( 1 + x^4 / (1 + x^2))) = 1 + x / (1 - x / (1 + x^3 / (1 + x^2 / (1 + x / (1 - x))))). - _Michael Somos_, Jan 03 2013

%F a(n) = A099443(n-1), n>0. - _R. J. Mathar_, Aug 05 2009

%e G.f. = 1 + x + x^2 + x^3 + x^4 - x^6 - x^7 - x^8 - x^9 + x^11 + x^12 + x^13 + ...

%t a[ n_] := Boole[n == 0] + (-1)^Quotient[n, 5] Sign@Mod[n, 5]; (* _Michael Somos_, Jun 17 2015 *)

%o (PARI) {a(n) = (n==0) + [0, 1, 1, 1, 1, 0, -1, -1, -1, -1][n%10 + 1]};

%Y Cf. A099443, A163811, A163818.

%K sign,easy

%O 0,1

%A _Michael Somos_, Aug 04 2009

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