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A010048
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Triangle of Fibonomial coefficients.
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41
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1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 3, 6, 3, 1, 1, 5, 15, 15, 5, 1, 1, 8, 40, 60, 40, 8, 1, 1, 13, 104, 260, 260, 104, 13, 1, 1, 21, 273, 1092, 1820, 1092, 273, 21, 1, 1, 34, 714, 4641, 12376, 12376, 4641, 714, 34, 1, 1, 55, 1870, 19635, 85085, 136136, 85085, 19635, 1870, 55, 1
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OFFSET
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0,8
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COMMENTS
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Conjecture: polynomials with (positive) Fibonomial coefficients are reducible iff n odd >1. - Ralf Stephan, Oct 29 2004
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REFERENCES
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Paul Barry, On Integer-Sequence-Based Constructions of Generalized Pascal Triangles, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.4.
A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, p. 15.
A. Brousseau, Fibonacci and Related Number Theoretic Tables. Fibonacci Association, San Jose, CA, 1972.
D. E. Knuth, The Art of Computer Programming. Addison-Wesley, Reading, MA, Vol. 1, p. 84 and 492.
A. T. Benjamin and S. S. Plott, A Combinatorial Approach to Fibonomial Coefficients, The Fibonacci Quarterly, Vol. 46/47, No. 1, pp. 7--9, February 2008/2009.
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LINKS
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T. D. Noe, Rows n=0..50 of triangle, flattened
E. Krot, An introduction to finite Fibonomial calculus
E. Krot, Further developments in Fibonomial calculus
T. M. Richardson, The Filbert Matrix, arXiv:math/9905079
R. Stephan, A recurrence for the fibonomials
Ron Knott The Fibonomials
Eric Weisstein's World of Mathematics, Fibonacci Coefficient
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FORMULA
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a(n, k) = ((n, k)) = F(n)*F(n-1)*...*F(n-k+1)/F(k)*F(k-1)*...*F(1), F(i) = Fibonacci numbers A000045.
a(n, k) = F(n-k-1)*a(n-1, k-1) + F(k+1)*a(n-1, k)
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EXAMPLE
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First few rows of the triangle a(n, k) are:
n\k 0 1 2 3 4 5 6 7 8 9 10
0: 1
1: 1 1
2: 1 1 1
3: 1 2 2 1
4: 1 3 6 3 1
5: 1 5 15 15 5 1
6: 1 8 40 60 40 8 1
7: 1 13 104 260 260 104 13 1
8: 1 21 273 1092 1820 1092 273 21 1
9: 1 34 714 4641 12376 12376 4641 714 34 1
10: 1 55 1870 19635 85085 136136 85085 19635 1870 55 1
... - Table extended and reformatted by Wolfdieter Lang, Oct 10 2012
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MATHEMATICA
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f[n_, k_] := Product[ Fibonacci[n - j + 1]/Fibonacci[j], {j, k}]; Table[ f[n, i], {n, 0, 10}, {i, 0, n}] [From Robert G. Wilson v, Dec 04 2009]
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PROG
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(Maxima) ffib(n):=prod(fib(k), k, 1, n);
fibonomial(n, k):=ffib(n)/(ffib(k)*ffib(n-k));
create_list(fibonomial(n, k), n, 0, 20, k, 0, n);
[Emanuele Munarini, Apr 2 2012]
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CROSSREFS
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Cf. A055870 (signed version of triangle). Row sums give A056569.
Columns include A000045, A001654, A001655, A001656, A001657, A001658, A056565, A056566, A056567.
Cf. A144712. [From Robert G. Wilson v, Dec 04 2009]
Sequence in context: A155865 A156133 * A055870 A088459 A007799 A122888
Adjacent sequences: A010045 A010046 A010047 * A010049 A010050 A010051
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KEYWORD
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nonn,tabl,easy,nice
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AUTHOR
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N. J. A. Sloane.
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STATUS
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approved
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