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Triangle, read by rows, T(n, k) = Fibonacci(n, k), where Fibonacci(n, x) is the Fibonacci polynomial.
6

%I #31 Oct 02 2024 14:27:56

%S 0,1,1,0,1,2,1,2,5,10,0,3,12,33,72,1,5,29,109,305,701,0,8,70,360,1292,

%T 3640,8658,1,13,169,1189,5473,18901,53353,129949,0,21,408,3927,23184,

%U 98145,328776,927843,2298912,1,34,985,12970,98209,509626,2026009,6624850,18674305,46866034

%N Triangle, read by rows, T(n, k) = Fibonacci(n, k), where Fibonacci(n, x) is the Fibonacci polynomial.

%D Steven Wolfram, The Mathematica Book, Cambridge University Press, 3rd ed. 1996, page 728

%H Alois P. Heinz, <a href="/A117715/b117715.txt">Rows n = 0..140, flattened</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/FibonacciPolynomial.html">Fibonacci Polynomial</a>.

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Fibonacci_polynomial">Fibonacci Polynomial</a>

%F T(n, 1) = A000045(n).

%F T(n, 3) = A006190(n).

%F T(n, 4) = A001076(n).

%F T(n, 5) = A052918(n-1).

%F T(5, k) = A057721(k).

%F T(6, k) = A124152(k).

%F T(n, k) = (-1)^(n-1)*A352361(n-k, n). - _G. C. Greubel_, Oct 01 2024

%e Triangle begins as:

%e 0;

%e 1, 1;

%e 0, 1, 2;

%e 1, 2, 5, 10;

%e 0, 3, 12, 33, 72;

%e 1, 5, 29, 109, 305, 701;

%e 0, 8, 70, 360, 1292, 3640, 8658;

%e 1, 13, 169, 1189, 5473, 18901, 53353, 129949;

%p with(combinat):for n from 0 to 9 do seq(fibonacci(n,m), m = 0 .. n) od; # _Zerinvary Lajos_, Apr 09 2008

%t Table[Fibonacci[n, k], {n,0,12}, {k,0,n}]//Flatten

%o (Python)

%o from sympy import fibonacci

%o def T(n, m): return 0 if n==0 else fibonacci(n, m)

%o for n in range(21): print([T(n, m) for m in range(n + 1)]) # _Indranil Ghosh_, Aug 12 2017

%o (Magma)

%o A117715:= func< n, k | k eq 0 select (n mod 2) else Evaluate(DicksonSecond(n-1, -1), k) >;

%o [A117715(n, k): k in [0..n], n in [0..13]]; // _G. C. Greubel_, Oct 01 2024

%o (SageMath)

%o def A117715(n,k): return lucas_number1(n, k, -1)

%o flatten([[A117715(n,k) for k in range(n+1)] for n in range(13)]) # _G. C. Greubel_, Oct 01 2024

%Y Cf. A000045, A117716, A049310, A073133, A157103 (antidiagonals).

%Y Main diagonal and first lower diagonal give: A084844, A084845.

%Y Cf. A352361.

%K nonn,easy,tabl

%O 0,6

%A _Roger L. Bagula_, Apr 13 2006

%E Definition simplified by the Assoc. Editors of the OEIS, Nov 17 2009