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A010048 Triangle of Fibonomial coefficients. 45
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,8

COMMENTS

Conjecture: polynomials with (positive) Fibonomial coefficients are reducible iff n odd >1. - Ralf Stephan, Oct 29 2004

REFERENCES

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.

LINKS

T. D. Noe, Rows n=0..50 of triangle, flattened

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, S. S. Plott, A combinatorial approach to fibonomial coefficients, Fib. Quart. 46/47 (1) (2008/9) 7-9.

A. Brousseau, Fibonacci and Related Number Theoretic Tables, Fibonacci Association, San Jose, CA, 1972.

M. Dziemianczuk, Cobweb Sequences Map, See sequence (4).2.

Tom Edgar and Michael Z. Spivey, Multiplicative functions, generalized binomial coefficients, and generalized Catalan numbers, Journal of Integer Sequences, Vol. 19 (2016), Article 16.1.6.

S. Falcon, On The Generating Functions of the Powers of the K-Fibonacci Numbers, Scholars Journal of Engineering and Technology (SJET), 2014; 2 (4C):669-675.

Dale Gerdemann, Golden Ratio Base Digit Patterns for Columns of the Fibonomial Triangle, "Another interesting pattern is for Golden Rectangle Numbers A001654. I made a short video illustrating this pattern, along with other columns of the Fibonomial Triangle A010048".

Ron Knott The Fibonomials

E. Krot, An introduction to finite Fibonomial calculus, arXiv:math/0503210 [math.CO], 2005.

E. Krot, Further developments in Fibonomial calculus, arXiv:math/0410550 [math.CO], 2004.

D. Marques, P. Trojovsky, On Divisibility of Fibonomial Coefficients by 3, J. Int. Seq. 15 (2012) #12.6.4

R. Mestrovic, Lucas' theorem: its generalizations, extensions and applications (1878--2014), arXiv preprint arXiv:1409.3820 [math.NT], 2014.

C. Pita, On s-Fibonomials, J. Int. Seq. 14 (2011) # 11.3.7

C. J. Pita Ruiz Velasco, Sums of Products of s-Fibonacci Polynomial Sequences, J. Int. Seq. 14 (2011) # 11.7.6

T. M. Richardson, The Filbert Matrix, arXiv:math/9905079 [math.RA], 1992.

Jeremiah Southwick, A Conjecture concerning the Fibonomial Triangle, arXiv:1604.04775 [math.NT], 2016.

R. Stephan, A recurrence for the fibonomials

Eric Weisstein's World of Mathematics, Fibonacci Coefficient, q-Binomial Coefficient.

FORMULA

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

EXAMPLE

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

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

MAPLE

A010048 := proc(n, k)

    mul(combinat[fibonacci](i), i=n-k+1..n)/mul(combinat[fibonacci](i), i=1..k) ;

end proc:

seq(seq(A010048(n, k), k=0..n), n=0..10) ; # R. J. Mathar, Feb 05 2015

MATHEMATICA

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 *)

PROG

(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 */

CROSSREFS

Cf. A055870 (signed version of triangle). Row sums give A056569.

Columns include A000045, A001654, A001655, A001656, A001657, A001658, A056565, A056566, A056567.

Cf. A144712. - Robert G. Wilson v, Dec 04 2009

Sequence in context: A155865 A156133 * A055870 A088459 A007799 A122888

Adjacent sequences:  A010045 A010046 A010047 * A010049 A010050 A010051

KEYWORD

nonn,tabl,easy,nice

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified November 18 10:57 EST 2017. Contains 294887 sequences.