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A010048
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Triangle of Fibonomial coefficients.
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48
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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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A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, p. 15.
D. E. Knuth, The Art of Computer Programming. Addison-Wesley, Reading, MA, Vol. 1, p. 84 and 492.
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LINKS
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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).
a(n, k) = phi^(k*(n-k)) * C(n, k)_{-1/phi^2}, where phi = (1+sqrt(5))/2 = A001622 is the golden ratio, and C(n, k)_q is the q-binomial coefficient. - Vladimir Reshetnikov, Sep 26 2016
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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
For n=7 and k=3, n - k + 1 = 7 - 3 + 1 = 5, so a(7,3) = F(7)*F(6)*F(5)/(F(3)*F(2)*F(1)) = 13*8*5/(2*1*1) = 520/2 = 260. - Michael B. Porter, Sep 26 2016
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MAPLE
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mul(combinat[fibonacci](i), i=n-k+1..n)/mul(combinat[fibonacci](i), i=1..k) ;
end proc:
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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}] (* Robert G. Wilson v, Dec 04 2009 *)
Column[Round@Table[GoldenRatio^(k(n-k)) QBinomial[n, k, -1/GoldenRatio^2], {n, 0, 10}, {k, 0, n}], Center] (* Round is equivalent to FullSimplify here, but is much faster - Vladimir Reshetnikov, Sep 25 2016 *)
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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 02 2012 */
(PARI) T(n, k) = prod(j=0, k-1, fibonacci(n-j))/prod(j=1, k, fibonacci(j));
tabl(nn) = for (n=0, nn, for (k=0, n, print1(T(n, k), ", ")); print); \\ Michel Marcus, Jul 20 2018
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CROSSREFS
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Cf. A055870 (signed version of triangle).
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KEYWORD
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AUTHOR
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STATUS
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approved
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