A081569 Fourth binomial transform of F(n+1).

1, 5, 26, 139, 757, 4172, 23165, 129217, 722818, 4050239, 22718609, 127512940, 715962889, 4020920141, 22584986378, 126867394723, 712691811325, 4003745802188, 22492567804517, 126361939999081, 709898671705906, 3988211185370615, 22405825905923321, 125876420631268204

Offset: 0

Author

Paul Barry, Mar 22 2003

Keywords

• easy
• nonn

Comments

Binomial transform of A081568.

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.

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

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200
• Edyta Hetmaniok, Bożena Piątek, and Roman Wituła, Binomials Transformation Formulae of Scaled Fibonacci Numbers, Open Mathematics, 15(1) (2017), 477-485.
• Roman Witula and Damian Slota, delta-Fibonacci numbers, Appl. Anal. Discr. Math 3 (2009), 310-329, MR2555042.
• Index entries for linear recurrences with constant coefficients, signature (9,-19).

Crossrefs

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

Programs

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

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

Maple

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

Mathematica

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

PARI

Vec((1-4*x)/(1-9*x+19*x^2) + O(x^30)) \\ Altug Alkan, Dec 10 2015

Sage

def A081569_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P((1-4*x)/(1-9*x+19*x^2)).list()
A081569_list(30) # G. C. Greubel, Aug 12 2019

Formula

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

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

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

a(n) = Sum_{k=0..n} A094441(n,k)*4^k. - Philippe Deléham, Dec 14 2009

a(n) = A081574(n) - 4*A081574(n-1). - R. J. Mathar, Jul 19 2012

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