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A081570 Fifth binomial transform of F(n+1).

%I A081570 #21 Sep 08 2022 08:45:09
%S A081570 1,6,37,233,1490,9633,62753,410926,2700349,17786985,117346714,
%T A081570 774991289,5121849473,33865596822,223987930325,1481764925737,
%U A081570 9803764203682,64870223394129,429263295428641,2840659771285310,18798621916707821
%N A081570 Fifth binomial transform of F(n+1).
%C A081570 Binomial transform of A081569.
%C A081570 Case k=5 of family of recurrences a(n) = (2k+1)*a(n-1) - A028387(k-1)*a(n-2), a(0)=1, a(1)=k+1.
%H A081570 Vincenzo Librandi, <a href="/A081570/b081570.txt">Table of n, a(n) for n = 0..200</a>
%H A081570 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (11,-29).
%F A081570 a(n) = 11*a(n-1) - 29*a(n-2), a(0)=1, a(1)=6.
%F A081570 a(n) = (1/2 - sqrt(5)/10)*(11/2 - sqrt(5)/2)^n + (sqrt(5)/10 + 1/2)*(sqrt(5)/2 + 11/2)^n .
%F A081570 G.f.: (1-5*x)/(1-11*x+29*x^2).
%F A081570 a(n) = Sum_{k=0..n} A094441(n,k)*5^k. - _Philippe Deléham_, Dec 14 2009
%p A081570 seq(coeff(series((1-5*x)/(1-11*x+29*x^2), x, n+1), x, n), n = 0 .. 30); # _G. C. Greubel_, Aug 12 2019
%t A081570 CoefficientList[Series[(1-5x)/(1 -11x +29x^2), {x, 0, 30}], x] (* _Vincenzo Librandi_, Aug 09 2013 *)
%t A081570 LinearRecurrence[{11,-29},{1,6},30] (* _Harvey P. Dale_, Aug 04 2022 *)
%o A081570 (Magma) I:=[1, 6]; [n le 2 select I[n] else 11*Self(n-1)-29*Self(n-2): n in [1..30]]; // _Vincenzo Librandi_, Aug 09 2013
%o A081570 (PARI) my(x='x+O('x^30)); Vec((1-5*x)/(1-11*x+29*x^2)) \\ _G. C. Greubel_, Aug 12 2019
%o A081570 (Sage)
%o A081570 def A081570_list(prec):
%o A081570     P.<x> = PowerSeriesRing(ZZ, prec)
%o A081570     return P((1-5*x)/(1-11*x+29*x^2)).list()
%o A081570 A081570_list(30) # _G. C. Greubel_, Aug 12 2019
%o A081570 (GAP) a:=[1,6];; for n in [3..30] do a[n]:=11*a[n-1]-29*a[n-2]; od; a; # _G. C. Greubel_, Aug 12 2019
%Y A081570 Cf. A000045.
%K A081570 easy,nonn
%O A081570 0,2
%A A081570 _Paul Barry_, Mar 22 2003