A322044 Triangle read by rows: numerators of coefficients (highest degree first) of polynomials interpolating Fibonacci numbers.

1, 1, 2, 1, 3, 6, 1, 3, 14, 30, 1, 2, 23, 94, 192, 1, 0, 35, 180, 744, 1560, 1, -3, 55, 255, 1744, 7308, 15120, 1, -7, 91, 245, 3304, 19922, 82284, 171360, 1, -12, 154, 0, 5929, 40572, 255996, 1068240, 2217600, 1, -18, 258, -756, 11361, 64638, 602972, 3746376, 15533568, 32296320

Offset: 0

Author

N. J. A. Sloane, Dec 07 2018

Keywords

• tabl
• sign

Comments

Row n has denominator n!.

Examples

			Triangle begins:
  1;
  1,  2;
  1,  3,  6;
  1,  3, 14,  30;
  1,  2, 23,  94,  192;
  1,  0, 35, 180,  744, 1560;
  1, -3, 55, 255, 1744, 7308, 15120;
  ...
		

References

• Brian Hopkins and Aram Tangboonduangjit, Fibonacci-producing rational polynomials, Fib. Q., 56:4 (2018), 303-312.

Links

• Alois P. Heinz, Rows n = 0..140, flattened (first 17 rows from Brian Hopkins)

Crossrefs

Main diagonal gives A078700(n+1).

Second column is negation of A167544.

Cf. A000045, A000142.

Programs

Maple

F:= proc(n) option remember; (<<0|1>, <1|1>>^n)[1, 2] end:
T:= n-> (p-> seq(coeff(p, x, n-j), j=0..n))(n!*expand(add(
      F(i+n+2)*binomial(x, i)*binomial(n-x, n-i), i=0..n))):
seq(T(n), n=0..10);  # Alois P. Heinz, Feb 24 2019

Mathematica

F[n_] := F[n] = MatrixPower[{{0, 1}, {1, 1}}, n][[1, 2]];
T[n_] := Function[p, Table[Coefficient[p, x, n - j], {j, 0, n}]][n! * FunctionExpand[Sum[F[i + n + 2] Binomial[x, i] Binomial[n - x, n - i], {i, 0, n}]]];
T /@ Range[0, 10] // Flatten (* Jean-François Alcover, May 29 2020, after Alois P. Heinz *)

Formula

The degree n polynomial is defined to be the interpolating polynomial of (0, F(n+2)), (1, F(n+3)), ..., (n,F(2n+2)) where F(n) is the n-th Fibonacci number. Theorem 2.1 of the paper proves the alternative form Sum_{i=0..n} F(i+n+2) * binomial(x,i) * binomial(n-x,n-i). - Brian Hopkins, Feb 24 2019

Extensions

Edited by Brian Hopkins, Feb 24 2019