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%I #141 Oct 23 2023 02:25:02
%S 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,
%T 104,260,260,104,13,1,1,21,273,1092,1820,1092,273,21,1,1,34,714,4641,
%U 12376,12376,4641,714,34,1,1,55,1870,19635,85085,136136,85085,19635,1870,55,1
%N Triangle of Fibonomial coefficients.
%C Conjecture: polynomials with (positive) Fibonomial coefficients are reducible iff n odd > 1. - _Ralf Stephan_, Oct 29 2004
%D A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, p. 15.
%D D. E. Knuth, The Art of Computer Programming. Addison-Wesley, Reading, MA, Vol. 1, p. 84 and 492.
%H T. D. Noe, <a href="/A010048/b010048.txt">Rows n = 0..50 of triangle, flattened</a>
%H Paul Barry, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL9/Barry/barry91.html">On Integer-Sequence-Based Constructions of Generalized Pascal Triangles</a>, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.4.
%H A. T. Benjamin and S. S. Plott, <a href="http://www.fq.math.ca/Papers1/46_47-1/Benjamin_11-08.pdf">A combinatorial approach to fibonomial coefficients</a>, Fib. Quart. 46/47 (1) (2008/9) 7-9.
%H A. Brousseau, <a href="http://www.fq.math.ca/fibonacci-tables.html">Fibonacci and Related Number Theoretic Tables</a>, Fibonacci Association, San Jose, CA, 1972.
%H Johann Cigler, <a href="https://arxiv.org/abs/2103.01652">Pascal triangle, Hoggatt matrices, and analogous constructions</a>, arXiv:2103.01652 [math.CO], 2021.
%H M. Dziemianczuk, <a href="http://www.faces-of-nature.art.pl/cobweb-sequences.html">Cobweb Sequences Map</a>, See sequence (4).2.
%H Tom Edgar and Michael Z. Spivey, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL19/Edgar/edgar3.html">Multiplicative functions, generalized binomial coefficients, and generalized Catalan numbers</a>, Journal of Integer Sequences, Vol. 19 (2016), Article 16.1.6.
%H P. F. F. Espinosa, J. F. González, J. P. Herrán, A. M. Cañadas, and J. L. Ramírez, <a href="https://doi.org/10.12958/adm1663">On some relationships between snake graphs and Brauer configuration algebras</a>, Algebra Disc. Math. (2022) Vol. 33, No. 2, 29-59.
%H S. Falcon, <a href="http://saspublisher.com/wp-content/uploads/2014/06/SJET24C669-675.pdf">On The Generating Functions of the Powers of the K-Fibonacci Numbers</a>, Scholars Journal of Engineering and Technology (SJET), 2014; 2 (4C):669-675.
%H Dale Gerdemann, <a href="https://www.youtube.com/watch?v=1LtjGV-nLG0">Golden Ratio Base Digit Patterns for Columns of the Fibonomial Triangle</a>, "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".
%H Dale K. Hathaway and Stephen L. Brown, <a href="https://digitalcommons.olivet.edu/math_facp/1/">Fibonacci Powers and a Fascinating Triangle</a>, The College Mathematics Journal, 28 (No. 2, 1997), 124-128. See Fig. 1.
%H Ron Knott, <a href="http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/Fibonomials.html">The Fibonomials</a>.
%H E. Krot, <a href="https://arxiv.org/abs/math/0503210">An introduction to finite Fibonomial calculus</a>, arXiv:math/0503210 [math.CO], 2005.
%H E. Krot, <a href="https://arxiv.org/abs/math/0410550">Further developments in Fibonomial calculus</a>, arXiv:math/0410550 [math.CO], 2004.
%H D. Marques and P. Trojovsky, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL15/Trojovsky/trojovsky2.html">On Divisibility of Fibonomial Coefficients by 3</a>, J. Int. Seq. 15 (2012) #12.6.4.
%H D. Marques and P. Trojovsky, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL18/Marques2/marques11.html">The p-adic order of some fibonomial coefficients</a>, J. Int. Seq. 18 (2015) # 15.3.1.
%H R. Mestrovic, <a href="http://arxiv.org/abs/1409.3820">Lucas' theorem: its generalizations, extensions and applications (1878--2014)</a>, arXiv preprint arXiv:1409.3820 [math.NT], 2014.
%H Phakhinkon Phunphayap, <a href="http://ithesis-ir.su.ac.th/dspace/bitstream/123456789/3040/1/59305804.pdf">Various Problems Concerning Factorials, Binomial Coefficients, Fibonomial Coefficients, and Palindromes</a>, Ph. D. Thesis, Silpakorn University (Thailand 2021).
%H Phakhinkon Phunphayap and Prapanpong Pongsriiam, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL21/Pongsriiam/pong12.html">Explicit Formulas for the p-adic Valuations of Fibonomial Coefficients</a>, J. Int. Seq. 21 (2018), #18.3.1.
%H C. Pita, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL14/Pita/pita12.html">On s-Fibonomials</a>, J. Int. Seq. 14 (2011) # 11.3.7.
%H C. J. Pita Ruiz Velasco, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL14/Pita2/pita8.html">Sums of Products of s-Fibonacci Polynomial Sequences</a>, J. Int. Seq. 14 (2011) # 11.7.6.
%H T. M. Richardson, <a href="http://arXiv.org/abs/math/9905079">The Filbert Matrix</a>, arXiv:math/9905079 [math.RA], 1992.
%H Bruce Sagan, <a href="https://users.math.msu.edu/users/bsagan/Slides/bca.pdf">Two Binomial Coefficient Analogues</a>, Slides, 2013.
%H Jeremiah Southwick, <a href="http://arxiv.org/abs/1604.04775">A Conjecture concerning the Fibonomial Triangle</a>, arXiv:1604.04775 [math.NT], 2016.
%H Ralf Stephan, <a href="/A010048/a010048conj.png">A recurrence for the fibonomials</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/FibonacciCoefficient.html">Fibonacci Coefficient</a>, <a href="http://mathworld.wolfram.com/q-BinomialCoefficient.html">q-Binomial Coefficient</a>.
%F 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.
%F a(n, k) = F(n-k-1)*a(n-1, k-1) + F(k+1)*a(n-1, k).
%F 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
%e First few rows of the triangle a(n, k) are:
%e n\k 0 1 2 3 4 5 6 7 8 9 10
%e 0: 1
%e 1: 1 1
%e 2: 1 1 1
%e 3: 1 2 2 1
%e 4: 1 3 6 3 1
%e 5: 1 5 15 15 5 1
%e 6: 1 8 40 60 40 8 1
%e 7: 1 13 104 260 260 104 13 1
%e 8: 1 21 273 1092 1820 1092 273 21 1
%e 9: 1 34 714 4641 12376 12376 4641 714 34 1
%e 10: 1 55 1870 19635 85085 136136 85085 19635 1870 55 1
%e ... - Table extended and reformatted by _Wolfdieter Lang_, Oct 10 2012
%e 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
%p A010048 := proc(n,k)
%p mul(combinat[fibonacci](i),i=n-k+1..n)/mul(combinat[fibonacci](i),i=1..k) ;
%p end proc:
%p seq(seq(A010048(n,k),k=0..n),n=0..10) ; # _R. J. Mathar_, Feb 05 2015
%t 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 *)
%t 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 *)
%o (Maxima) ffib(n):=prod(fib(k),k,1,n);
%o fibonomial(n,k):=ffib(n)/(ffib(k)*ffib(n-k));
%o create_list(fibonomial(n,k),n,0,20,k,0,n); /* _Emanuele Munarini_, Apr 02 2012 */
%o (PARI) T(n, k) = prod(j=0, k-1, fibonacci(n-j))/prod(j=1, k, fibonacci(j));
%o tabl(nn) = for (n=0, nn, for (k=0, n, print1(T(n, k), ", ")); print); \\ _Michel Marcus_, Jul 20 2018
%Y Cf. A055870 (signed version of triangle).
%Y Row sums give A056569.
%Y Columns include A000045, A001654, A001655, A001656, A001657, A001658, A056565, A056566, A056567.
%Y Cf. A144712. - _Robert G. Wilson v_, Dec 04 2009
%Y Cf. A181926 (antidiagonal sums), A181927 (row square-sums), A003267 and A003268 (central Fibonomial coefficients), A003150 (Fibonomial Catalan numbers).
%K nonn,tabl,easy,nice
%O 0,8
%A _N. J. A. Sloane_
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