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%I #37 Sep 08 2022 08:45:13
%S 1,1,1,1,3,3,3,3,7,7,8,8,14,14,16,16,25,25,29,29,41,41,47,47,63,63,72,
%T 72,92,92,104,104,129,129,145,145,175,175,195,195,231,231,256,256,298,
%U 298,328,328,377,377,413,413,469,469,511,511,575,575,624,624,696,696,752,752
%N Expansion of (1+x^10)/((1-x)*(1-x^4)^2*(1-x^8)).
%H G. Nebe, E. M. Rains and N. J. A. Sloane, <a href="http://neilsloane.com/doc/cliff2.html">Self-Dual Codes and Invariant Theory</a>, Springer, Berlin, 2006.
%H <a href="/index/Mo#Molien">Index entries for Molien series</a>
%H <a href="/index/Rec#order_15">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,-1,1,-1,-1,1,1,-1,-1,1,-1,1,1,-1).
%F G.f.: ( 1+x^4+x^8-x^2-x^6 ) / ( (1+x^4) * (1+x^2)^2 * (1+x)^3 * (1-x)^4 ). - _R. J. Mathar_, Dec 18 2014
%F a(n) = (2*n^3 + 21*n^2 + 175*n + 441 + 3*(n^2 + 7*n + 29)*(-1)^n + 30*(2*n + 7)*(-1)^((2*n - 1 + (-1)^n)/4) + 30*(-1)^((6*n - 1 + (-1)^n)/4) + 48*((-1)^((2*n - 3 + (-1)^n + 2*(-1)^((2*n - 1 + (-1)^n)/4))/8) - (-1)^((6*n - 5 + 3*(-1)^n + 2*(-1)^((2*n - 1 + (-1)^n)/4))/8)))/768. - _Luce ETIENNE_, Mar 31 2015
%F Shorter version of above:
%F a(n) = (2*n^3 + 21*n^2 + 175*n + 441 + 3*(n^2 + 7*n + 29)*(-1)^n + 30*(2*n + 7)*sign(1-n%4+n%2) + 30*sign(2-(n+1)%4-n%2) + 48*((5*n-n^2)%4)*sign(5-n%8))/768, where sign(x) = x/abs(x), and a%b = a (mod b). - _Derek Orr_, Apr 05 2015
%t CoefficientList[Series[(1 + x^4 + x^8 - x^2 - x^6) / ((x^4 + 1) (1 + x^2)^2 (1 + x)^3 (x-1)^4), {x, 0, 70}], x] (* _Vincenzo Librandi_, Apr 05 2015 *)
%o (PARI) Vec((1+x^10)/((1-x)*(1-x^4)^2*(1-x^8)) + O(x^80)) \\ _Michel Marcus_, Apr 05 2015
%o (PARI)
%o a(n) = (2*n^3 + 21*n^2 + 175*n + 441 + 3*(n^2 + 7*n + 29)*(-1)^n + 30*(2*n + 7)*sign(1-n%4+n%2) + 30*sign(2-(n+1)%4-n%2) + 48*((5*n-n^2)%4)*sign(5-n%8))/768
%o vector(100,n,a(n-1)) \\ _Derek Orr_, Apr 05 2015
%o (Magma) m:=70; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!((1+x^4+x^8-x^2-x^6)/((1+x^4)*(1+x^2)^2*(1+x)^3*(1-x)^4))); // _Bruno Berselli_, Apr 07 2015
%K nonn,easy
%O 0,5
%A _N. J. A. Sloane_, Apr 08 2004
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