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

%I A214178 #24 Jan 05 2025 19:51:39
%S A214178 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,
%T A214178 13,38,102,240,480,720,720,0,21,71,222,630,1560,3240,5040,5040,0,34,
%U A214178 130,466,1536,4560,11760,25200,40320,40320,0,55,235,948,3564,12264
%N A214178 Triangle T(n,k) by rows: the k-th derivative of the Fibonacci Polynomial F_n(x) evaluated at x=1.
%C A214178 T(n,0) = A000045(n), Fibonacci numbers;
%C A214178 T(n,1) = A001629(n) for n > 0;
%C A214178 T(n,n-3) = A038720(n-2) for n > 2;
%C A214178 T(n,n-2) = A000142(n-1) for n > 1;
%C A214178 T(n,n-1) = A000142(n-1) for n > 0;
%C A214178 T(n,n) = 0.
%H A214178 Reinhard Zumkeller, <a href="/A214178/b214178.txt">Rows n = 0..150 of triangle, flattened</a>
%H A214178 P. Filipponi, A. F. Horadam, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/31-3.html">Second derivative sequences of Fibonacci and Lucas Polynomials</a>, Fib. Quart. 31 (1993), 194-204.
%F A214178 T(n,k) = A037027(n,k)*k!, 0 <= k < n; T(n,n) = 0.
%e A214178 The triangle begins:
%e A214178 .   0: [0]
%e A214178 .   1: [1, 0]
%e A214178 .   2: [1, 1, 0]
%e A214178 .   3: [2, 2, 2, 0]
%e A214178 .   4: [3, 5, 6, 6, 0]
%e A214178 .   5: [5, 10, 18, 24, 24, 0]
%e A214178 .   6: [8, 20, 44, 84, 120, 120, 0]
%e A214178 .   7: [13, 38, 102, 240, 480, 720, 720, 0]
%e A214178 .   8: [21, 71, 222, 630, 1560, 3240, 5040, 5040, 0]
%e A214178 .   9: [34, 130, 466, 1536, 4560, 11760, 25200, 40320, 40320, 0]
%e A214178 .  10: [55, 235, 948, 3564, 12264, 37800, 100800, 221760, 362880, 362880, 0]
%e A214178        ...
%t A214178 T[n_, k_] := D[Fibonacci[n, x], {x, k}] /. x -> 1;
%t A214178 Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Sep 20 2021 *)
%o A214178 (Haskell)
%o A214178 a214178 n k = a214178_tabl !! n !! k
%o A214178 a214178_row n = a214178_tabl !! n
%o A214178 a214178_tabl = [0] : map f a037027_tabl where
%o A214178    f row = (zipWith (*) a000142_list row) ++ [0]
%Y A214178 Cf. A000142, A037027.
%K A214178 nonn,tabl
%O A214178 0,7
%A A214178 _R. J. Mathar_, _L. Edson Jeffery_, _Reinhard Zumkeller_, Jul 07 2012