A099133 -id:A099133 - cp's OEIS frontend

A234357 Array T(n,k) by antidiagonals: T(n,k) = n^k * Fibonacci(k).

1, 2, 2, 3, 8, 3, 4, 18, 24, 5, 5, 32, 81, 80, 8, 6, 50, 192, 405, 256, 13, 7, 72, 375, 1280, 1944, 832, 21, 8, 98, 648, 3125, 8192, 9477, 2688, 34, 9, 128, 1029, 6480, 25000, 53248, 45927, 8704, 55, 10, 162, 1536, 12005, 62208, 203125, 344064, 223074, 28160, 89, 11, 200, 2187

Offset: 0

Ralf Stephan, Dec 24 2013

			Array starts:
1,  2,   3,    5,     8,     13,    21,   34, 55, 89,...    (A000045)
2,  8,  24,   80,   256,    832,  2688, 8704,...   (A063727, A085449)
3, 18,  81,  405,  1944,   9477, 45927,...         (A122069, A099012)
4, 32, 192, 1280,  8192,  53248,...                         (A099133)
5, 50, 375, 3125, 25000, 203125,...
6, 72, 648, 6480, 62208, 606528,...
...
Columns: A000027, A001105, A117642.

PARI
```
T(n,k)=n^k*fibonacci(k)
```

T(n,k)=polcoeff(Ser(1/(1-n*x-n^2*x^2)),k)

G.f. of n-th row: 1/(1 - n*x - n^2*x^2).

Recurrence: T(n,k) = n*T(n,k-1) + n^2*T(n,k-2), starting n, 2*n^2.

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

Original entry on oeis.org

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

Offset: 0

Paul Barry, Sep 29 2004

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

R. Witula, D. Slota, \delta-Fibonacci Numbers, Appl. Anal. Discrete Math., 3 (2009), 310-329.

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

Cf. A015447.

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 *)

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}.

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A234357 Array T(n,k) by antidiagonals: T(n,k) = n^k * Fibonacci(k).

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

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