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A082289 Expansion of x^4*(2+x)/((1+x)*(1-x)^5).

%I A082289 #29 Mar 15 2024 12:50:55
%S A082289 2,9,26,59,116,206,340,530,790,1135,1582,2149,2856,3724,4776,6036,
%T A082289 7530,9285,11330,13695,16412,19514,23036,27014,31486,36491,42070,
%U A082289 48265,55120,62680,70992,80104,90066,100929,112746,125571,139460,154470
%N A082289 Expansion of x^4*(2+x)/((1+x)*(1-x)^5).
%H A082289 Vincenzo Librandi, <a href="/A082289/b082289.txt">Table of n, a(n) for n = 4..10000</a>
%H A082289 <a href="/index/Tu#2wis">Index entries for two-way infinite sequences</a>
%H A082289 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (4,-5,0,5,-4,1).
%F A082289 G.f.: x^4*(2+x)/((1+x)*(1-x)^5).
%F A082289 a(n) = 3*a(n-1) - 2*a(n-2) - 2*a(n-3) + 3*a(n-4) - a(n-5) + 3. If sequence is also defined for n <= 3 by this equation, then a(n)=0 for 0 <= n <= 3 and a(n) = A070893(-n) for n < 0.
%F A082289 a(n) = A082290(2*n-7).
%F A082289 a(n) = (1/96)*(2*(n-2)*n*(3*n^2 - 10*n + 4) + 3*(-1)^n - 3). a(n) - a(n-2) = A006002(n-3) for n > 5. - _Bruno Berselli_, Aug 26 2011
%F A082289 a(n) = 4*a(n-1) - 5*a(n-2) + 5*a(n-4) - 4*a(n-5) + a(n-6); a(4)=2, a(5)=9, a(6)=26, a(7)=59, a(8)=116, a(9)=206. - _Harvey P. Dale_, Aug 26 2013
%t A082289 Drop[CoefficientList[Series[x^4(2+x)/((1+x)(1-x)^5),{x,0,50}],x],4] (* or *) LinearRecurrence[{4,-5,0,5,-4,1},{2,9,26,59,116,206},50] (* _Harvey P. Dale_, Aug 26 2013 *)
%o A082289 (PARI) a(n)=polcoeff(if(n>0,x^4*(2+x)/((1+x)*(1-x)^5),x*(1+2*x)/((1+x)*(1-x)^5))+x*O(x^abs(n)),abs(n))
%o A082289 (Magma) [(1/96)*(2*(n-2)*n*(3*n^2-10*n+4)+3*(-1)^n-3): n in [4..50]]; // _Vincenzo Librandi_, Aug 29 2011
%Y A082289 Cf. A070893, A082290.
%Y A082289 Cf. A045947 (which contains the first differences). - _Bruno Berselli_, Aug 26 2011
%K A082289 nonn,easy
%O A082289 4,1
%A A082289 _Michael Somos_, Apr 07 2003