A099134 Expansion of x/(1-2x-19x^2).

0, 1, 2, 23, 84, 605, 2806, 17107, 87528, 500089, 2663210, 14828111, 80257212, 442248533, 2409384094, 13221490315, 72221278416, 395650872817, 2163506035538, 11844378654599, 64795371984420, 354633938406221

Offset: 0

Author

Paul Barry, Sep 29 2004

Keywords

• easy
• nonn

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

• Index entries for linear recurrences with constant coefficients, signature (2,19).

Crossrefs

Cf. A015447.

Programs

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 *)

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}.