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%I #40 Oct 17 2022 01:45:31
%S 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,
%T 1744,7308,15120,1,-7,91,245,3304,19922,82284,171360,1,-12,154,0,5929,
%U 40572,255996,1068240,2217600,1,-18,258,-756,11361,64638,602972,3746376,15533568,32296320
%N Triangle read by rows: numerators of coefficients (highest degree first) of polynomials interpolating Fibonacci numbers.
%C Row n has denominator n!.
%D Brian Hopkins and Aram Tangboonduangjit, Fibonacci-producing rational polynomials, Fib. Q., 56:4 (2018), 303-312.
%H Alois P. Heinz, <a href="/A322044/b322044.txt">Rows n = 0..140, flattened</a> (first 17 rows from Brian Hopkins)
%F 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
%e Triangle begins:
%e 1;
%e 1, 2;
%e 1, 3, 6;
%e 1, 3, 14, 30;
%e 1, 2, 23, 94, 192;
%e 1, 0, 35, 180, 744, 1560;
%e 1, -3, 55, 255, 1744, 7308, 15120;
%e ...
%p F:= proc(n) option remember; (<<0|1>, <1|1>>^n)[1, 2] end:
%p T:= n-> (p-> seq(coeff(p, x, n-j), j=0..n))(n!*expand(add(
%p F(i+n+2)*binomial(x, i)*binomial(n-x, n-i), i=0..n))):
%p seq(T(n), n=0..10); # _Alois P. Heinz_, Feb 24 2019
%t F[n_] := F[n] = MatrixPower[{{0, 1}, {1, 1}}, n][[1, 2]];
%t 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 T /@ Range[0, 10] // Flatten (* _Jean-François Alcover_, May 29 2020, after _Alois P. Heinz_ *)
%Y Main diagonal gives A078700(n+1).
%Y Second column is negation of A167544.
%Y Cf. A000045, A000142.
%K tabl,sign
%O 0,3
%A _N. J. A. Sloane_, Dec 07 2018
%E Edited by _Brian Hopkins_, Feb 24 2019
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