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

0,3

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).

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

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 *)

Prog

(Sage) [lucas_number1(n, 9, 19) for n in range(0, 21)] # Zerinvary Lajos, Apr 23 2009
(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
(PARI) my(x='x+O('x^30)); Vec(x/(1 - 9*x + 19*x^2)) \\ G. C. Greubel, Aug 13 2019
(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

Crossrefs

Cf. A000045, A081569.

Sequence in context: A328955 A098921 A027234 * A084151 A346396 A240391

Adjacent sequences: A081571 A081572 A081573 * A081575 A081576 A081577

Keyword

easy,nonn

Author

Paul Barry, Mar 22 2003

Extensions

Corrected by Philippe Deléham, Dec 16 2009

Status

approved