%I #46 Sep 08 2022 08:45:10
%S 1,4,19,94,469,2344,11719,58594,292969,1464844,7324219,36621094,
%T 183105469,915527344,4577636719,22888183594,114440917969,572204589844,
%U 2861022949219,14305114746094,71525573730469,357627868652344
%N 4th row of number array A083064.
%C Inverse binomial transform of A090040. [_Paul Curtz_, Jan 11 2009]
%C Let A be the Hessenberg matrix of order n, defined by: A[1,j]=1, A[i,i]:=7, (i>1), A[i,i-1]=-1, and A[i,j]=0 otherwise. Then, for n>=1, a(n-1)=(-1)^(n-1)*charpoly(A,2). [_Milan Janjic_, Feb 21 2010]
%C For an integer x, consider the sequence P(x) of polynomials p_{1}, p_{2}, p_{3}, . . . defined by p_{1} = x-1, p_{n+1} = x*p_{1} - 1. P(5) = This sequence. P(1), P(2), P(3), P(4) are A000004, A123412, A007051, A007583 resp. [_K.V.Iyer_, Jun 22 2010]
%C It appears that if s(n) is a first order rational sequence of the form s(0)=2, s(n)= (3*s(n-1)+2)/(2*s(n-1)+3), n>0, then s(n)=2*a(n)/(2*a(n)-1), n>0.
%C An Engel expansion of 5/3 to the base b := 5/4 as defined in A181565, with the associated series expansion 5/3 = b + b^2/4 + b^3/(4*19) + b^4/(4*19*94) + .... Cf. A007051. - _Peter Bala_, Oct 29 2013
%H Vincenzo Librandi, <a href="/A083065/b083065.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (6,-5).
%F a(n) = (3*5^n+1)/4.
%F G.f.: (1-2*x)/((1-5*x)(1-x)).
%F E.g.f.: (3*exp(5*x) + exp(x))/4.
%F a(n) = 5*a(n-1)-1 with n>0, a(0)=1. - _Vincenzo Librandi_, Aug 08 2010
%F a(n) = 6*a(n-1)-5*a(n-2). - _Vincenzo Librandi_, Nov 04 2011
%F a(n) = 5^n - Sum_{i=0..n-1} 5^i. - _Bruno Berselli_, Jun 20 2013
%p a[0]:=0:a[1]:=1:for n from 2 to 50 do a[n]:=a[n-1]*5-1 od: seq(a[n], n=1..22); # _Zerinvary Lajos_, Feb 22 2008
%t CoefficientList[Series[(1-2x)/((1-5x)(1-x)),{x,0,30}],x] (* or *) LinearRecurrence[{6,-5},{1,4},30] (* _Harvey P. Dale_, Jul 27 2022 *)
%o (Magma) [(3*5^n+1)/4: n in [0..30]]; // _Vincenzo Librandi_, Nov 04 2011
%Y Cf. A007583, A083066, A007051.
%K nonn,easy
%O 0,2
%A _Paul Barry_, Apr 21 2003