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A322044 Triangle read by rows: numerators of coefficients (highest degree first) of polynomials interpolating Fibonacci numbers. 1
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 (list; table; graph; refs; listen; history; text; internal format)
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 as 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-Fran├žois 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

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Last modified September 16 21:28 EDT 2021. Contains 347473 sequences. (Running on oeis4.)