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

0,5

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.

Links

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

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)

Example

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;

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 *)

Prog

(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
(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

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.

Sequence in context: A038137 A073133 A106179 * A292630 A144287 A106196

Adjacent sequences: A081569 A081570 A081571 * A081573 A081574 A081575

Keyword

easy,nonn,tabl

Author

Paul Barry, Mar 22 2003

Status

approved