A214178 Triangle T(n,k) by rows: the k-th derivative of the Fibonacci Polynomial F_n(x) evaluated at x=1.

0, 1, 0, 1, 1, 0, 2, 2, 2, 0, 3, 5, 6, 6, 0, 5, 10, 18, 24, 24, 0, 8, 20, 44, 84, 120, 120, 0, 13, 38, 102, 240, 480, 720, 720, 0, 21, 71, 222, 630, 1560, 3240, 5040, 5040, 0, 34, 130, 466, 1536, 4560, 11760, 25200, 40320, 40320, 0, 55, 235, 948, 3564, 12264

Offset: 0

Author

R. J. Mathar, L. Edson Jeffery, Reinhard Zumkeller, Jul 07 2012

Keywords

• nonn
• tabl

Comments

T(n,0) = A000045(n), Fibonacci numbers;

T(n,1) = A001629(n) for n > 0;

T(n,n-3) = A038720(n-2) for n > 2;

T(n,n-2) = A000142(n-1) for n > 1;

T(n,n-1) = A000142(n-1) for n > 0;

T(n,n) = 0.

Examples

			The triangle begins:
.   0: [0]
.   1: [1, 0]
.   2: [1, 1, 0]
.   3: [2, 2, 2, 0]
.   4: [3, 5, 6, 6, 0]
.   5: [5, 10, 18, 24, 24, 0]
.   6: [8, 20, 44, 84, 120, 120, 0]
.   7: [13, 38, 102, 240, 480, 720, 720, 0]
.   8: [21, 71, 222, 630, 1560, 3240, 5040, 5040, 0]
.   9: [34, 130, 466, 1536, 4560, 11760, 25200, 40320, 40320, 0]
.  10: [55, 235, 948, 3564, 12264, 37800, 100800, 221760, 362880, 362880, 0]
       ...
		

Links

• Reinhard Zumkeller, Rows n = 0..150 of triangle, flattened
• P. Filipponi, A. F. Horadam, Second derivative sequences of Fibonacci and Lucas Polynomials, Fib. Quart. 31 (1993), 194-204.

Crossrefs

Cf. A000142, A037027.

Programs

Haskell

a214178 n k = a214178_tabl !! n !! k
a214178_row n = a214178_tabl !! n
a214178_tabl = [0] : map f a037027_tabl where
   f row = (zipWith (*) a000142_list row) ++ [0]

Mathematica

T[n_, k_] := D[Fibonacci[n, x], {x, k}] /. x -> 1;
Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Sep 20 2021 *)

Formula

T(n,k) = A037027(n,k)*k!, 0 <= k < n; T(n,n) = 0.