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

0,3

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

Table of n, a(n) for n=0..21.

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

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

Cf. A015447.

Sequence in context: A097232 A339474 A222564 * A069152 A131464 A245331

Adjacent sequences: A099131 A099132 A099133 * A099135 A099136 A099137

Keyword

easy,nonn

Author

Paul Barry, Sep 29 2004

Status

approved