OFFSET
0,3
COMMENTS
Binomial transform is A099133. Binomial transform of x/(1-20x^2), or (0,1,0,20,0,400,0,8000,....). The inverse binomial transform of k^(n-1)Fib(n) has g.f. x/(1-(k-2)x-(k^2+k-1)x^2).
4*a(n) = (-1)^(n+1)*b(n;4) = 3^n*b(n;4/3), where b(n;d), n=0,1,..., d \in C, denote one of the delta-Fibonacci numbers defined in comments to A014445 (see also Witula-Slota's paper). Our first identity is equivalent to the second formula given below. We note that the sequence (4/3)^n*F(n) is the binomial transform of the sequence 3^(-n)*b(n;4). - Roman Witula, Jul 24 2012
REFERENCES
R. Witula, D. Slota, \delta-Fibonacci Numbers, Appl. Anal. Discrete Math., 3 (2009), 310-329.
LINKS
FORMULA
a(n) = 2a(n-1) + 19a(n-2).
a(n) = sum{k=0..n, (-1)^(n-k)binomial(n, k)4^(k-1)*Fib(k)}.
a(n) = sum{k=0..n, binomial(n, 2k+1)20^k}.
MATHEMATICA
Join[{a=0, b=1}, Table[c=2*b+19*a; a=b; b=c, {n, 60}]] (* Vladimir Joseph Stephan Orlovsky, Feb 01 2011 *)
CoefficientList[Series[x/(1-2x-19x^2), {x, 0, 30}], x] (* or *) LinearRecurrence[ {2, 19}, {0, 1}, 30] (* Harvey P. Dale, Dec 25 2019 *)
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Paul Barry, Sep 29 2004
STATUS
approved