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A081574 Fourth binomial transform of Fibonacci numbers F(n). 7
0, 1, 9, 62, 387, 2305, 13392, 76733, 436149, 2467414, 13919895, 78398189, 441105696, 2480385673, 13942462833, 78354837710, 440286745563, 2473838793577, 13899100976496, 78088971710501, 438717826841085 (list; graph; refs; listen; history; text; internal format)
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

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 - 9 x + 19 x^2), {x, 0,  30}], x] (* Vincenzo Librandi, Aug 09 2013 *)

PROG

(Sage) [lucas_number1(n, 9, 19) for n in xrange(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

CROSSREFS

Cf. A000045, A081569.

Sequence in context: A264376 A098921 A027234 * A084151 A240391 A229701

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

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Last modified July 21 17:20 EDT 2019. Contains 325198 sequences. (Running on oeis4.)