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

Author

Paul Barry, Mar 22 2003

Keywords

• easy
• nonn
• tabl

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.

Examples

			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;
		

Links

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

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.

Programs

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

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 *)

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

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)