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

0,3

Comments

Row n has denominator n!.

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)

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

Example

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;
  ...

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 *)

Crossrefs

Main diagonal gives A078700(n+1).

Second column is negation of A167544.

Cf. A000045, A000142.

Sequence in context: A238955 A238968 A217891 * A010251 A051537 A338797

Adjacent sequences: A322041 A322042 A322043 * A322045 A322046 A322047

Keyword

tabl,sign

Author

N. J. A. Sloane, Dec 07 2018

Extensions

Edited by Brian Hopkins, Feb 24 2019

Status

approved