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

0,6

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

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.

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

Example

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,
...

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 *)

Prog

(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

Crossrefs

Diagonals include A000045, A259709, A006502.

Cf. A000142 (row sums).

Sequence in context: A265017 A349136 A035376 * A029220 A249901 A253274

Adjacent sequences: A259705 A259706 A259707 * A259709 A259710 A259711

Keyword

sign,tabl,easy

Author

N. J. A. Sloane, Jul 05 2015

Extensions

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

Status

approved