%I #28 Dec 21 2024 23:51:15

%S 1,1,0,-2,-4,-4,1,15,43,91,166,276,430,638,911,1261,1701,2245,2908,

%T 3706,4656,5776,7085,8603,10351,12351,14626,17200,20098,23346,26971,

%U 31001,35465,40393,45816,51766,58276,65380,73113,81511,90611,100451,111070,122508

%N a(n) = binomial(n,0) - binomial(n,2) + binomial(n,4).

%H Harry J. Smith, <a href="/A058923/b058923.txt">Table of n, a(n) for n = 0..200</a>

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).

%F From _R. J. Mathar_, Mar 17 2009: (Start)

%F a(n) = 5a(n-1)-10a(n-2)+10a(n-3)-5a(n-4)+a(n-5).

%F G.f.: -(1-4*x-2*x^3+x^4+5*x^2)/(x-1)^5. (End)

%t CoefficientList[Series[-((z - 1)*z*((z - 1)*z + 4) + 1)/(z - 1)^5, {z, 0, 100}], z] (* _Vladimir Joseph Stephan Orlovsky_, Jul 16 2011 *)

%t Table[Binomial[n,0]-Binomial[n,2]+Binomial[n,4],{n,0,50}] (* or *) LinearRecurrence[{5,-10,10,-5,1},{1,1,0,-2,-4},50] (* _Harvey P. Dale_, Mar 02 2015 *)

%o (PARI) a(n) = 1 - binomial(n,2) + binomial(n,4) \\ _Harry J. Smith_, Jun 24 2009

%o (PARI) Vec(-(1-4*x-2*x^3+x^4+5*x^2)/(x-1)^5 + O(x^60)) \\ _Michel Marcus_, Jan 03 2016

%Y Cf. A000127.

%K sign,easy

%O 0,4

%A _N. J. A. Sloane_, Jan 12 2001