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A117715 Triangle, read by rows, T(n, k) = Fibonacci(n, k), where Fibonacci(n, x) is the Fibonacci polynomial.

%I A117715 #32 Feb 16 2025 08:33:00
%S A117715 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 A117715 3640,8658,1,13,169,1189,5473,18901,53353,129949,0,21,408,3927,23184,
%U A117715 98145,328776,927843,2298912,1,34,985,12970,98209,509626,2026009,6624850,18674305,46866034
%N A117715 Triangle, read by rows, T(n, k) = Fibonacci(n, k), where Fibonacci(n, x) is the Fibonacci polynomial.
%D A117715 Steven Wolfram, The Mathematica Book, Cambridge University Press, 3rd ed. 1996, page 728
%H A117715 Alois P. Heinz, <a href="/A117715/b117715.txt">Rows n = 0..140, flattened</a>
%H A117715 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/FibonacciPolynomial.html">Fibonacci Polynomial</a>.
%H A117715 Wikipedia, <a href="http://en.wikipedia.org/wiki/Fibonacci_polynomial">Fibonacci Polynomial</a>
%F A117715 T(n, 1) = A000045(n).
%F A117715 T(n, 3) = A006190(n).
%F A117715 T(n, 4) = A001076(n).
%F A117715 T(n, 5) = A052918(n-1).
%F A117715 T(5, k) = A057721(k).
%F A117715 T(6, k) = A124152(k).
%F A117715 T(n, k) = (-1)^(n-1)*A352361(n-k, n). - _G. C. Greubel_, Oct 01 2024
%e A117715 Triangle begins as:
%e A117715   0;
%e A117715   1,  1;
%e A117715   0,  1,   2;
%e A117715   1,  2,   5,   10;
%e A117715   0,  3,  12,   33,   72;
%e A117715   1,  5,  29,  109,  305,   701;
%e A117715   0,  8,  70,  360, 1292,  3640,  8658;
%e A117715   1, 13, 169, 1189, 5473, 18901, 53353, 129949;
%p A117715 with(combinat):for n from 0 to 9 do seq(fibonacci(n,m), m = 0 .. n) od; # _Zerinvary Lajos_, Apr 09 2008
%t A117715 Table[Fibonacci[n, k], {n,0,12}, {k,0,n}]//Flatten
%o A117715 (Python)
%o A117715 from sympy import fibonacci
%o A117715 def T(n, m): return 0 if n==0 else fibonacci(n, m)
%o A117715 for n in range(21): print([T(n, m) for m in range(n + 1)]) # _Indranil Ghosh_, Aug 12 2017
%o A117715 (Magma)
%o A117715 A117715:= func< n, k | k eq 0 select (n mod 2) else Evaluate(DicksonSecond(n-1, -1), k) >;
%o A117715 [A117715(n, k): k in [0..n], n in [0..13]]; // _G. C. Greubel_, Oct 01 2024
%o A117715 (SageMath)
%o A117715 def A117715(n,k): return lucas_number1(n, k, -1)
%o A117715 flatten([[A117715(n,k) for k in range(n+1)] for n in range(13)]) # _G. C. Greubel_, Oct 01 2024
%Y A117715 Cf. A000045, A117716, A049310, A073133, A157103 (antidiagonals).
%Y A117715 Main diagonal and first lower diagonal give: A084844, A084845.
%Y A117715 Cf. A352361.
%K A117715 nonn,easy,tabl
%O A117715 0,6
%A A117715 _Roger L. Bagula_, Apr 13 2006
%E A117715 Definition simplified by the Assoc. Editors of the OEIS, Nov 17 2009