A081572 Square array of binomial transforms of Fibonacci numbers, read by ascending antidiagonals.

1, 1, 1, 1, 2, 2, 1, 3, 5, 3, 1, 4, 10, 13, 5, 1, 5, 17, 35, 34, 8, 1, 6, 26, 75, 125, 89, 13, 1, 7, 37, 139, 338, 450, 233, 21, 1, 8, 50, 233, 757, 1541, 1625, 610, 34, 1, 9, 65, 363, 1490, 4172, 7069, 5875, 1597, 55, 1, 10, 82, 535, 2669, 9633, 23165, 32532, 21250, 4181, 89

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) = 1 and a(1) = k+1.

Examples

			The array rows begins as:
  1, 1,  2,   3,    5,     8,     13, ... A000045;
  1, 2,  5,  13,   34,    89,    233, ... A001519;
  1, 3, 10,  35,  125,   450,   1625, ... A081567;
  1, 4, 17,  75,  338,  1541,   7069, ... A081568;
  1, 5, 26, 139,  757,  4172,  23165, ... A081569;
  1, 6, 37, 233, 1490,  9633,  62753, ... A081570;
  1, 7, 50, 363, 2669, 19814, 148153, ... A081571;
Antidiagonal triangle begins as:
  1;
  1, 1;
  1, 2,  2;
  1, 3,  5,   3;
  1, 4, 10,  13,   5;
  1, 5, 17,  35,  34,    8;
  1, 6, 26,  75, 125,   89,   13;
  1, 7, 37, 139, 338,  450,  233,  21;
  1, 8, 50, 233, 757, 1541, 1625, 610, 34;
		

Links

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

Crossrefs

Array row n: A000045 (n=0), A001519 (n=1), A081567 (n=2), A081568 (n=3), A081569 (n=4), A081570 (n=5), A081571 (n=6).

Array column k: A000027 (k=1), A002522 (k=2).

Different from A073133.

Cf. A028387.

Programs

Magma

A081572:= func< n,k | (&+[Binomial(k,j)*Fibonacci(j+1)*(n-k)^(k-j): j in [0..k]]) >;
[A081572(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, May 27 2021

Mathematica

T[n_, k_]:= If[n==0, Fibonacci[k+1], Sum[Binomial[k, j]*Fibonacci[j+1]*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 A081572(n,k): return sum( binomial(k,j)*fibonacci(j+1)*(n-k)^(k-j) for j in (0..k) )
flatten([[A081572(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, May 27 2021

Formula

Rows are successive binomial transforms of F(n+1).

T(n, k) = ((5+sqrt(5))/10)*( (2*n + 1 + sqrt(5))/2)^k + ((5-sqrt(5)/10)*( 2*n + 1 - sqrt(5))/2 )^k.

From G. C. Greubel, May 27 2021: (Start)

T(n, k) = Sum_{j=0..k} binomial(k,j)*n^(k-j)*Fibonacci(j+1) (square array).

T(n, k) = Sum_{j=0..k} binomial(k,j)*(n-k)^(k-j)*Fibonacci(j+1) (antidiagonal triangle). (End)