This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A081574 #38 Sep 08 2022 08:45:09
%S A081574 0,1,9,62,387,2305,13392,76733,436149,2467414,13919895,78398189,
%T A081574 441105696,2480385673,13942462833,78354837710,440286745563,
%U A081574 2473838793577,13899100976496,78088971710501,438717826841085
%N A081574 Fourth binomial transform of Fibonacci numbers F(n).
%C A081574 Binomial transform of A099453(n-1):= [0,1,7,38,189,905,...].
%C A081574 Case k=4 of family of recurrences a(n) = (2k+1)*a(n-1) - A028387(k-1)*a(n-2), a(0)=0, a(1)=1.
%H A081574 Vincenzo Librandi, <a href="/A081574/b081574.txt">Table of n, a(n) for n = 0..200</a>
%H A081574 S. Falcon, <a href="http://dx.doi.org/10.9734/BJMCS/2014/11783">Iterated Binomial Transforms of the k-Fibonacci Sequence</a>, British Journal of Mathematics & Computer Science, 4 (22): 2014.
%H A081574 J. Pan, <a href="https://cs.uwaterloo.ca/journals/JIS/OL13/Pan/pan8.html">Multiple Binomial Transforms and Families of Integer Sequences </a>, J. Int. Seq. 13 (2010), 10.4.2, F^(4).
%H A081574 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (9,-19).
%F A081574 a(n) = 9*a(n-1) - 19*a(n-1), a(0)=0, a(1)=1.
%F A081574 a(n) = ((sqrt(5)/2 + 9/2)^n - (9/2 - sqrt(5)/2)^n)/sqrt(5).
%F A081574 G.f.: x/(1 - 9*x + 19*x^2).
%F A081574 E.g.f.: 2*exp(9*x/2)*sinh(sqrt(5)*x/2)/sqrt(5). - _Ilya Gutkovskiy_, Aug 11 2017
%p A081574 seq(coeff(series(x/(1-9*x+19*x^2), x, n+1), x, n), n = 0..30); # _G. C. Greubel_, Aug 13 2019
%t A081574 Join[{a=0,b=1},Table[c=9*b-19*a;a=b;b=c,{n,60}]] (* _Vladimir Joseph Stephan Orlovsky_, Jan 27 2011 *)
%t A081574 LinearRecurrence[{9,-19},{0,1},30] (* _Harvey P. Dale_, Dec 03 2011 *)
%t A081574 CoefficientList[Series[x/(1 -9x +19x^2), {x, 0, 30}], x] (* _Vincenzo Librandi_, Aug 09 2013 *)
%o A081574 (Sage) [lucas_number1(n,9,19) for n in range(0, 21)] # _Zerinvary Lajos_, Apr 23 2009
%o A081574 (Magma) [n le 2 select (n-1) else 9*Self(n-1)-19*Self(n-2): n in [1..25]]; // _Vincenzo Librandi_, Aug 09 2013
%o A081574 (PARI) my(x='x+O('x^30)); Vec(x/(1 - 9*x + 19*x^2)) \\ _G. C. Greubel_, Aug 13 2019
%o A081574 (GAP) a:=[0,1];; for n in [3..30] do a[n]:=9*a[n-1]-19*a[n-2]; od; a; # _G. C. Greubel_, Aug 13 2019
%Y A081574 Cf. A000045, A081569.
%K A081574 easy,nonn
%O A081574 0,3
%A A081574 _Paul Barry_, Mar 22 2003
%E A081574 Corrected by _Philippe Deléham_, Dec 16 2009