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
KEYWORD
AUTHOR
Paul Barry, Mar 22 2003
STATUS
approved