A081574 Fourth binomial transform of Fibonacci numbers F(n).

0, 1, 9, 62, 387, 2305, 13392, 76733, 436149, 2467414, 13919895, 78398189, 441105696, 2480385673, 13942462833, 78354837710, 440286745563, 2473838793577, 13899100976496, 78088971710501, 438717826841085

Offset: 0

Author

Paul Barry, Mar 22 2003

Keywords

• easy
• nonn

Comments

Binomial transform of A099453(n-1):= [0,1,7,38,189,905,...].

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.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200
• S. Falcon, Iterated Binomial Transforms of the k-Fibonacci Sequence, British Journal of Mathematics & Computer Science, 4 (22): 2014.
• J. Pan, Multiple Binomial Transforms and Families of Integer Sequences , J. Int. Seq. 13 (2010), 10.4.2, F^(4).
• Index entries for linear recurrences with constant coefficients, signature (9,-19).

Crossrefs

Cf. A000045, A081569.

Programs

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

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

Maple

seq(coeff(series(x/(1-9*x+19*x^2), x, n+1), x, n), n = 0..30); # G. C. Greubel, Aug 13 2019

Mathematica

Join[{a=0,b=1},Table[c=9*b-19*a;a=b;b=c,{n,60}]] (* Vladimir Joseph Stephan Orlovsky, Jan 27 2011 *)
LinearRecurrence[{9,-19},{0,1},30] (* Harvey P. Dale, Dec 03 2011 *)
CoefficientList[Series[x/(1 -9x +19x^2), {x, 0,  30}], x] (* Vincenzo Librandi, Aug 09 2013 *)

PARI

my(x='x+O('x^30)); Vec(x/(1 - 9*x + 19*x^2)) \\ G. C. Greubel, Aug 13 2019

Sage

[lucas_number1(n,9,19) for n in range(0, 21)] # Zerinvary Lajos, Apr 23 2009

Formula

a(n) = 9*a(n-1) - 19*a(n-1), a(0)=0, a(1)=1.

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

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

E.g.f.: 2*exp(9*x/2)*sinh(sqrt(5)*x/2)/sqrt(5). - Ilya Gutkovskiy, Aug 11 2017

Extensions

Corrected by Philippe Deléham, Dec 16 2009