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

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, 3640, 8658, 1, 13, 169, 1189, 5473, 18901, 53353, 129949, 0, 21, 408, 3927, 23184, 98145, 328776, 927843, 2298912, 1, 34, 985, 12970, 98209, 509626, 2026009, 6624850, 18674305, 46866034

Offset: 0

Author

Roger L. Bagula, Apr 13 2006

Keywords

• nonn
• easy
• tabl

Examples

			Triangle begins as:
  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,  3640,  8658;
  1, 13, 169, 1189, 5473, 18901, 53353, 129949;
		

References

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

Links

• Alois P. Heinz, Rows n = 0..140, flattened
• Eric Weisstein's World of Mathematics, Fibonacci Polynomial.
• Wikipedia, Fibonacci Polynomial

Crossrefs

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

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

Cf. A352361.

Programs

Magma

A117715:= func< n, k | k eq 0 select (n mod 2) else Evaluate(DicksonSecond(n-1, -1), k) >;
[A117715(n, k): k in [0..n], n in [0..13]]; // G. C. Greubel, Oct 01 2024

Maple

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

Mathematica

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

Python

from sympy import fibonacci
def T(n, m): return 0 if n==0 else fibonacci(n, m)
for n in range(21): print([T(n, m) for m in range(n + 1)]) # Indranil Ghosh, Aug 12 2017

SageMath

def A117715(n,k): return lucas_number1(n, k, -1)
flatten([[A117715(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Oct 01 2024

Formula

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

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

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

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

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

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

T(n, k) = (-1)^(n-1)*A352361(n-k, n). - G. C. Greubel, Oct 01 2024

Extensions

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