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A117715
Triangle, read by rows, T(n, k) = Fibonacci(n, k), where Fibonacci(n, x) is the Fibonacci polynomial.
6
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
0,6
REFERENCES
Steven Wolfram, The Mathematica Book, Cambridge University Press, 3rd ed. 1996, page 728
LINKS
Eric Weisstein's World of Mathematics, Fibonacci Polynomial.
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
EXAMPLE
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;
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
PROG
(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
(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
(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
CROSSREFS
Cf. A000045, A117716, A049310, A073133, A157103 (antidiagonals).
Main diagonal and first lower diagonal give: A084844, A084845.
Cf. A352361.
Sequence in context: A199599 A201163 A049901 * A330962 A327194 A160457
KEYWORD
nonn,easy,tabl
AUTHOR
Roger L. Bagula, Apr 13 2006
EXTENSIONS
Definition simplified by the Assoc. Editors of the OEIS, Nov 17 2009
STATUS
approved