A081576 Square array of binomial transforms of Fibonacci numbers, read by antidiagonals.

0, 0, 1, 0, 1, 1, 0, 1, 3, 2, 0, 1, 5, 8, 3, 0, 1, 7, 20, 21, 5, 0, 1, 9, 38, 75, 55, 8, 0, 1, 11, 62, 189, 275, 144, 13, 0, 1, 13, 92, 387, 905, 1000, 377, 21, 0, 1, 15, 128, 693, 2305, 4256, 3625, 987, 34, 0, 1, 17, 170, 1131, 4955, 13392, 19837, 13125, 2584, 55

Offset

0,9

Comments

Array rows are solutions of the recurrence a(n) = (2*k+1)*a(n-1) - A028387(k-1)*a(n-2) where a(0) = 0 and a(1) = 1.

Links

G. C. Greubel, Antidiagonal rows n = 0..50, flattened

Formula

Rows are successive binomial transforms of F(n).

T(n, k) = ( ( (2*n + 1 + sqrt(5))/2 )^k - ( (2*n + 1 - sqrt(5))/2 )^k )/sqrt(5).

From G. C. Greubel, May 26 2021: (Start)

T(n, k) = Sum_{j=0..k} binomial(k,j)*Fibonacci(j)*n^(k-j) with T(0, k) = Fibonacci(k) (square array).

T(n, k) = Sum_{j=0..k} binomial(k,j)*Fibonacci(j)*(n-k)^(k-j) (antidiagonal triangle). (End)

Example

Square array begins as:
  0, 1,  1,   2,    3,    5,     8, ... A000045;
  0, 1,  3,   8,   21,   55,   144, ... A001906;
  0, 1,  5,  20,   75,  275,  1000, ... A030191;
  0, 1,  7,  38,  189,  905,  4256, ... A099453;
  0, 1,  9,  62,  387, 2305, 13392, ... A081574;
  0, 1, 11,  92,  693, 4955, 34408, ... A081575;
  0, 1, 13, 128, 1131, 9455, 76544, ...
The antidiagonal triangle begins as:
  0;
  0, 1;
  0, 1,  1;
  0, 1,  3,  2;
  0, 1,  5,  8,   3;
  0, 1,  7, 20,  21,   5;
  0, 1,  9, 38,  75,  55,   8;
  0, 1, 11, 62, 189, 275, 144, 13;

Mathematica

T[n_, k_]:= If[n==0, Fibonacci[k], Sum[Binomial[k, j]*Fibonacci[j]*n^(k-j), {j, 0, k}]]; Table[T[n-k, k], {n, 0, 12}, {k, 0, n}] //Flatten (* G. C. Greubel, May 26 2021 *)

Prog

(Magma)
A081576:= func< n, k | (&+[Binomial(k, j)*Fibonacci(j)*(n-k)^(k-j): j in [0..k]]) >;
[A081576(n, k): k in [0..n], n in [0..12]]; // G. C. Greubel, May 26 2021
(Sage)
def A081576(n, k): return sum( binomial(k, j)*fibonacci(j)*(n-k)^(k-j) for j in (0..k) )
flatten([[A081576(n, k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, May 26 2021

Crossrefs

Array row n: A000045 (n=0), A001906 (n=1), A030191 (n=2), A099453 (n=3), A081574 (n=4), A081575 (n=5).

Array columns k: A005408 (k=3), A077588 (k=4).

Cf. A028387, A081573, A081574, A081575.

Sequence in context: A139144 A360866 A373451 * A330785 A292717 A365727

Adjacent sequences: A081573 A081574 A081575 * A081577 A081578 A081579

Keyword

easy,nonn,tabl

Author

Paul Barry, Mar 22 2003

Status

approved