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Fourth binomial transform of F(n+1).
9

%I #56 Jun 03 2024 16:08:57

%S 1,5,26,139,757,4172,23165,129217,722818,4050239,22718609,127512940,

%T 715962889,4020920141,22584986378,126867394723,712691811325,

%U 4003745802188,22492567804517,126361939999081,709898671705906,3988211185370615,22405825905923321,125876420631268204

%N Fourth binomial transform of F(n+1).

%C Binomial transform of A081568.

%C Case k = 4 of family of recurrences a(n) = (2*k+1)*a(n-1) - A028387(k-1)*a(n-2) for n >= 2, with a(0) = 1 and a(1) = k + 1.

%C a(n) = 5^n * a(n;1/5) = Sum_{k=0..n} binomial(n,k) * (-1)^k * F(k-1) * 5^(n-k), which implies also Deléham's formula given below and where a(n;d), n=0,1,...,d, denote the delta-Fibonacci numbers defined in comments to A000045 (see also Witula's et al. papers). - _Roman Witula_, Jul 12 2012

%H Vincenzo Librandi, <a href="/A081569/b081569.txt">Table of n, a(n) for n = 0..200</a>

%H Edyta Hetmaniok, Bożena Piątek, and Roman Wituła, <a href="https://doi.org/10.1515/math-2017-0047">Binomials Transformation Formulae of Scaled Fibonacci Numbers</a>, Open Mathematics, 15(1) (2017), 477-485.

%H Roman Witula and Damian Slota, <a href="http://dx.doi.org/10.2298/AADM0902310W">delta-Fibonacci numbers</a>, Appl. Anal. Discr. Math 3 (2009), 310-329, <a href="http://www.ams.org/mathscinet-getitem?mr=2555042">MR2555042</a>.

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (9,-19).

%F a(n) = 9*a(n-1) - 19*a(n-2) for n >= 2, with a(0) = 1 and a(1) = 5.

%F a(n) = (1/2 - sqrt(5)/10)*(9/2 - sqrt(5)/2)^n + (sqrt(5)/10 + 1/2)*(sqrt(5)/2 + 9/2)^n.

%F G.f.: (1 - 4*x)/(1 - 9*x + 19*x^2).

%F a(n) = Sum_{k=0..n} A094441(n,k)*4^k. - _Philippe Deléham_, Dec 14 2009

%F a(n) = A081574(n) - 4*A081574(n-1). - _R. J. Mathar_, Jul 19 2012

%F E.g.f.: exp(9*x/2)*(5*cosh(sqrt(5)*x/2) + sqrt(5)*sinh(sqrt(5)*x/2))/5. - _Stefano Spezia_, Jun 03 2024

%p seq(coeff(series((1-4*x)/(1-9*x+19*x^2), x, n+1), x, n), n = 0 .. 30); # _G. C. Greubel_, Aug 12 2019

%t CoefficientList[Series[(1-4x)/(1 -9x +19x^2), {x, 0, 30}], x] (* _Vincenzo Librandi_, Aug 09 2013 *)

%o (Magma) I:=[1, 5]; [n le 2 select I[n] else 9*Self(n-1)-19*Self(n-2): n in [1..30]]; // _Vincenzo Librandi_, Aug 09 2013

%o (PARI) Vec((1-4*x)/(1-9*x+19*x^2) + O(x^30)) \\ _Altug Alkan_, Dec 10 2015

%o (Sage)

%o def A081569_list(prec):

%o P.<x> = PowerSeriesRing(ZZ, prec)

%o return P((1-4*x)/(1-9*x+19*x^2)).list()

%o A081569_list(30) # _G. C. Greubel_, Aug 12 2019

%o (GAP) a:=[1,5];; for n in [3..30] do a[n]:=9*a[n-1]-19*a[n-2]; od; a; # _G. C. Greubel_, Aug 12 2019

%Y Cf. A000045, A028387, A081568, A081574, A094441.

%K easy,nonn

%O 0,2

%A _Paul Barry_, Mar 22 2003