A259708 Triangle T(n,k) (0 <= k <= n) giving coefficients of certain polynomials related to Fibonacci numbers.

1, 0, 1, 1, -1, 2, 0, 3, 0, 3, 1, 0, 14, 4, 5, 0, 8, 22, 60, 22, 8, 1, 6, 99, 244, 279, 78, 13, 0, 21, 240, 1251, 2016, 1251, 240, 21, 1, 25, 715, 5245, 14209, 14083, 5329, 679, 34, 0, 55, 1828, 21532, 88060, 139930, 88060, 21532, 1828, 55, 1, 78, 4817, 83060, 507398, 1218920, 1219382, 507068, 83225, 4762, 89

Offset: 0

Author

N. J. A. Sloane, Jul 05 2015

Keywords

• sign
• tabl
• easy

Comments

The terms are the coefficients of the polynomials given by r_0(x) = 1; r_1(x) = x; r_(n+1) = (n+1)*x*r_n(x) + x*(1-x)*(r_n)'(x) + (1 - x)^2*r_(n-1)(x). [Carlitz, (1.6)]. Note: Carlitz wrongly states r_1(x) = 1. - Eric M. Schmidt, Jul 10 2015

Examples

			Triangle begins:
1,
0,1,
1,-1,2,
0,3,0,3,
1,0,14,4,5,
0,8,22,60,22,8,
1,6,99,244,279,78,13,
0,21,240,1251,2016,1251,240,21,
...
		

Links

• Eric M. Schmidt, Rows n = 0..50, flattened
• L. Carlitz, Some polynomials related to Fibonacci and Eulerian numbers, Fib. Quart., 16 (1978), 217. (Annotated scanned copy)
• L. Carlitz, Some polynomials related to Fibonacci and Eulerian numbers, Fib. Quart., 16 (1978), 216-226.

Crossrefs

Diagonals include A000045, A259709, A006502.

Cf. A000142 (row sums).

Programs

Maple

A259708  := proc(n,k)
    if k < 0 or k > n then
        0;
    elif k =0 and n =0 then
        1;
    else
        (n-k+1)*procname(n-1,k-1)+k*procname(n-1,k)+procname(n-2,k)-2*procname(n-2,k-1) + procname(n-2,k-2) ;
    end if ;
end proc: # R. J. Mathar, Jun 18 2019

Mathematica

T[n_, k_] := T[n, k] = If[k < 0 || k > n, 0, If[k == 0 && n == 0, 1, (n - k + 1) T[n - 1, k - 1] + k T[n - 1, k] + T[n - 2, k] - 2 T[n - 2, k - 1] + T[n - 2, k - 2]]];
Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Mar 30 2020 *)

Sage

@CachedFunction
def T(n,k) :
    if n < 0 or k < 0 : return 0
    if n == 0 and k == 0 : return 1
    return (n-k+1)*T(n-1,k-1) + k*T(n-1,k) + T(n-2,k) - 2*T(n-2,k-1) + T(n-2,k-2)
# Eric M. Schmidt, Jul 10 2015

Formula

T(0,0) = 1; T(n+1,k) = (n-k+2)*T(n,k-1) + k*T(n,k) + T(n-1,k) - 2*T(n-1,k-1) + T(n-1,k-2), where we put T(n,k) = 0 if n < 0 or k < 0. As special cases, T(n,n) = Fibonacci(n+1) and T(n,0) = 1 (n even) or 0 (n odd). - Rewritten by Eric M. Schmidt, Jul 10 2015

Extensions

More terms from and name revised by Eric M. Schmidt, Jul 10 2015