A056565 -id:A056565 - cp's OEIS frontend

A010048 Triangle of Fibonomial coefficients, read by rows.

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

Offset: 0

N. J. A. Sloane

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

			First few rows of the triangle T(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 T(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

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.

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 and 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.
Johann Cigler, Pascal triangle, Hoggatt matrices, and analogous constructions, arXiv:2103.01652 [math.CO], 2021.
M. Dziemianczuk, Cobweb Sequences Map, See sequence (4).2. [Dead link]
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.
P. F. F. Espinosa, J. F. González, J. P. Herrán, A. M. Cañadas, and J. L. Ramírez, On some relationships between snake graphs and Brauer configuration algebras, Algebra Disc. Math. (2022) Vol. 33, No. 2, 29-59.
Sergio 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".
Dale K. Hathaway and Stephen L. Brown, Fibonacci Powers and a Fascinating Triangle, The College Mathematics Journal, 28 (No. 2, 1997), 124-128. See Fig. 1.
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 and P. Trojovsky, On Divisibility of Fibonomial Coefficients by 3, J. Int. Seq. 15 (2012) #12.6.4.
D. Marques and P. Trojovsky, The p-adic order of some fibonomial coefficients, J. Int. Seq. 18 (2015) # 15.3.1.
Romeo Mestrovic, Lucas' theorem: its generalizations, extensions and applications (1878--2014), arXiv preprint arXiv:1409.3820 [math.NT], 2014.
Phakhinkon Phunphayap, Various Problems Concerning Factorials, Binomial Coefficients, Fibonomial Coefficients, and Palindromes, Ph. D. Thesis, Silpakorn University (Thailand 2021).
Phakhinkon Phunphayap and Prapanpong Pongsriiam, Explicit Formulas for the p-adic Valuations of Fibonomial Coefficients, J. Int. Seq. 21 (2018), #18.3.1.
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.
Bruce Sagan, Two Binomial Coefficient Analogues, Slides, 2013.
Jeremiah Southwick, A Conjecture concerning the Fibonomial Triangle, arXiv:1604.04775 [math.NT], 2016.
Ralf Stephan, A recurrence for the fibonomials.
Eric Weisstein's World of Mathematics, Fibonacci Coefficient, q-Binomial Coefficient.

Cf. A055870 (signed version of triangle).
Columns include: A000045, A001654, A001655, A001656, A001657, A001658, A056565, A056566, A056567.
Sums include: A056569 (row), A181926 (antidiagonal), A181927 (row square-sums).
Cf. A003267 and A003268 (central Fibonomial coefficients), A003150 (Fibonomial Catalan numbers), A144712, A099927, A385732/A385733 (Lucas).

Fibonomial:= func< n,k | k eq 0 select 1 else (&*[Fibonacci(n-j+1)/Fibonacci(j): j in [1..k]]) >;
[Fibonomial(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jul 20 2024

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

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 *)
T[n_, k_] := With[{c = ArcCsch[2] - I Pi/2}, Product[I^j Sinh[c j], {j, k + 1, n}] / Product[I^j Sinh[c j], {j, 1, n - k}]]; Table[Simplify[T[n, k]], {n, 0, 10}, {k, 0, n}] // Flatten  (* Peter Luschny, Jul 08 2025 *)

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 */

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

def fibonomial(n,k): return 1 if k==0 else product(fibonacci(n-j+1)/fibonacci(j) for j in range(1,k+1))
flatten([[fibonomial(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Jul 20 2024

T(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.
T(n, k) = Fibonacci(n-k-1)*T(n-1, k-1) + Fibonacci(k+1)*T(n-1, k).
T(n, k) = phi^(k*(n-k)) * C(n, k)A001622%20is%20the%20golden%20ratio,%20and%20C(n,%20k)_q%20is%20the%20q-binomial%20coefficient.%20-%20_Vladimir%20Reshetnikov">{-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
G.f. of column k: x^k * exp( Sum_{j>=1} Fibonacci((k+1)*j)/Fibonacci(j) * x^j/j ). - Seiichi Manyama, May 07 2025
T(n, k) = Product_{j=k+1..n} i^j*sinh(c*j) / Product_{j=1..n-k} i^j*sinh(c*j) where c = arccsch(2) - i*Pi/2 and i is the imaginary unit. If you substitute sinh by cosh you get the Lucas triangle A385732/A385733, which is a rational triangle. - Peter Luschny, Jul 08 2025

A055870 Signed Fibonomial triangle.

Original entry on oeis.org

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

Offset: 0

Wolfdieter Lang, Jul 10 2000

Row n+1 (n >= 1) of the signed triangle lists the coefficients of the recursion relation for the n-th power of Fibonacci numbers A000045: Sum_{m=0..n+1} T(n+1,m)*(Fibonacci(k-m))^n = 0, k >= n+1; inputs: (Fibonacci(k))^n, k=0..n.
The inverse of the row polynomial p(n,x) := Sum_{m=0..n} T(n,m)*x^m is the g.f. for the column m=n-1 of the Fibonomial triangle A010048.
The row polynomials p(n,x) factorize according to p(n,x) = G(n-1)*p(n-2,-x), with inputs p(0,x)= 1, p(1,x)= 1-x and G(n):= 1 - A000032(n)*x + (-1)^n*x^2. (Derived from Riordan's result and Knuth's exercise).
The row polynomials are the characteristic polynomials of product of the binomial matrix binomial(i,j) and the exchange matrix J_n (matrix with 1's on the antidiagonal, 0 elsewhere). - Paul Barry, Oct 05 2004

			Row polynomial for n=4: p(4,x) = 1-3*x-6*x^2+3*x^3+x^4 = (1+x-x^2)*(1-4*x-x^2). 1/p(4,x) is G.f. for A010048(n+3,3), n >= 0: {1,3,15,60,...} = A001655(n).
For n=3: 1*(Fibonacci(k))^3 - 3*(Fibonacci(k-1))^3 - 6*(Fibonacci(k-2))^3 + 3*(Fibonacci(k-3))^3 + 1*(Fibonacci(k-4))^3 = 0, k >= 4; inputs: (Fibonacci(k))^3, k=0..3.
The triangle begins:
  n\m 0   1     2    3     4      5     6    7   8   9
  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
  ... [_Wolfdieter Lang_, Aug 06 2012; a(7,1) corrected, Oct 10 2012]

D. E. Knuth, The Art of Computer Programming. Addison-Wesley, Reading, MA, 1969, Vol. 1, pp. 84-5 and 492.

G. C. Greubel, Rows n = 0..50 of the triangle, flattened
Katharine A. Ahrens, Combinatorial Applications of the k-Fibonacci Numbers: A Cryptographically Motivated Analysis, Ph. D. thesis, North Carolina State University (2020).
A. T. Benjamin, S. S. Plott, A combinatorial approach to fibonomial coefficients, Fib. Quart. 46/47 (1) (2008/9) 7-9.
A. Brousseau, A sequence of power formulas, Fib. Quart., 6 (1968), 81-83.
H. W. Gould, Extensions of the Hermite g.c.d. theorems for binomial coefficients, Fib Quart. 33 (1995) 386.
E. Kilic, The generalized Fibonomial matrix, Eur. J. Combinat. 31 (1) (2010) 193-209.
Ron Knott, The Fibonomials
Ewa Krot, An introduction to finite fibonomial calculus, Centr. Eur. J. Math. 2 (5) (2004) 754.
A. K. Kwasniewski, Fibonomial cumulative connection constants, arXiv:math/0406006 [math.CO], 2004-2009.
Phakhinkon Phunphayap, Various Problems Concerning Factorials, Binomial Coefficients, Fibonomial Coefficients, and Palindromes, Ph. D. Thesis, Silpakorn University (Thailand 2021).
Phakhinkon Phunphayap, Prapanpong Pongsriiam, Explicit Formulas for the p-adic Valuations of Fibonomial Coefficients, J. Int. Seq. 21 (2018), #18.3.1.
J. Riordan, Generating functions for powers of Fibonacci numbers, Duke. Math. J. 29 (1962) 5-12.
J. Seibert and P. Trojovsky, On some identities for the Fibonomial coefficients, Math. Slov. 55 (2005) 9-19.
P. Trojovsky, On some identities for the Fibonomial coefficients..., Discr. Appl. Math. 155 (15) (2007) 2017.

Cf. A000032, A000045, A001654, A001655, A001656, A001657, A001658, A010048, A051159, A056565, A056566, A056567.
Sums include: A055871 (signed row), A056569 (row).
Central column: A003268.
Cf. A383715.

Fibonomial:= func< n,k | k eq 0 select 1 else (&*[Fibonacci(n-j+1)/Fibonacci(j): j in [1..k]]) >;
[(-1)^Floor((k+1)/2)*Fibonomial(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jul 20 2024

A055870 := proc(n,k)
    (-1)^floor((k+1)/2)*A010048(n,k) ;
end proc: # R. J. Mathar, Jun 14 2015

T[n_, m_]:= {1,-1,-1,1}[[Mod[m,4] + 1]] * Product[ Fibonacci[n-j+1]/Fibonacci[j], {j, m}];
Table[T[n, m], {n, 0, 10}, {m, 0, n}]//Flatten (* Jean-François Alcover, Jul 05 2013 *)

def fibonomial(n,k): return 1 if k==0 else product(fibonacci(n-j+1)/fibonacci(j) for j in range(1,k+1))
flatten([[(-1)^((k+1)//2)*fibonomial(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Jul 20 2024

T(n, m) = (-1)^floor((m+1)/2)*A010048(n, m), where A010048(n, m) := fibonomial(n, m).
G.f. for column m: (-1)^floor((m+1)/2)*x^m/p(m+1, x) with the row polynomial of the (signed) triangle: p(n, x) := Sum_{m=0..n} T(n, m)*x^m.
Sum_{k=0..n} T(n,k) * x^k = exp( -Sum_{k>=1} Fibonacci(n*k)/Fibonacci(k) * x^k/k ). - Seiichi Manyama, May 07 2025

A056566 Fibonomial coefficients.

Original entry on oeis.org

1, 34, 1870, 83215, 3994320, 186135312, 8771626578, 411591708660, 19344810307020, 908637119420910, 42689423937884208, 2005443612183077232, 94214069697350815795, 4426039514623184676790, 207929935924379904006970, 9768275694729434277258589, 458901121999204061365680096

Offset: 0

Wolfdieter Lang, Jul 10 2000

Vincenzo Librandi, Table of n, a(n) for n = 0..100

Cf. A010048, A000045, A001654-8, A056565, A001906 (signed), A004187, A049660 (signed), A049668.

a[n_] := (1/65520) Times @@ Fibonacci[n + Range[8]]; Array[a, 20, 0] (* Giovanni Resta, May 08 2016 *)

b(n, k)=prod(j=1, k, fibonacci(n+j)/fibonacci(j));
vector(20, n, b(n-1, 8)) \\ Joerg Arndt, May 08 2016

a(n) = A010048(n+8, 8) = Fibonomial(n+8, 8).
G.f.: 1/p(9, n) with p(9, n)= 1 - 34*x - 714*x^2 + 4641*x^3 + 12376*x^4 - 12376*x^5 - 4641*x^6 + 714*x^7 + 34*x^8 - x^9 = (1-x)*(1 + 3*x + x^2)*(1 - 7*x + x^2)* (1 + 18*x + x^2)*(1 - 47*x + x^2) (n=9 row polynomial of signed Fibonomial triangle A055870; see this entry for Knuth and Riordan references).
Recursion: a(n) = 47*a(n-1) - a(n-2) + ((-1)^n)*A001658(n), n >= 2, a(0)=1, a(1)=34.
G.f.: exp( Sum_{k>=1} F(9*k)/F(k) * x^k/k ), where F(n) = A000045(n). - Seiichi Manyama, May 07 2025

A056567 Fibonomial coefficients.

Original entry on oeis.org

1, 55, 4895, 352440, 27372840, 2063912136, 157373300370, 11948265189630, 908637119420910, 69056421075989160, 5249543573067466872, 399024295188779925720, 30331388438447118520355, 2305576054220330112077285, 175254358052498673797874685, 13321629629800423781409595728

Offset: 0

Wolfdieter Lang, Jul 10 2000

Cf. A010048, A000045, A001654-8, A056565-6, A001076 (signed), A049666, A049667 (signed), A049669.

a[n_] := (1/2227680) Times @@ Fibonacci[n + Range[9]]; Array[a, 20, 0] (* Giovanni Resta, May 08 2016 *)

b(n, k)=prod(j=1, k, fibonacci(n+j)/fibonacci(j));
vector(20, n, b(n-1, 9))  \\ Joerg Arndt, May 08 2016

a(n) = A010048(n+9, 9) = Fibonomial(n+9, 9).
G.f.: 1/p(10, n) with p(10, n)= 1 - 55*x - 1870*x^2 + 19635*x^3 + 85085*x^4 - 136136*x^5 - 85085*x^6 + 19635*x^7 + 1870*x^8 - 55*x^9 - x^10 = (1 - x - x^2)*(1 + 4*x - x^2)*(1 - 11*x - x^2)*(1 + 29*x - x^2)*(1 - 76*x - x^2) (n=10 row polynomial of signed Fibonomial triangle A055870; see this entry for Knuth and Riordan references).
Recursion: a(n) = 76*a(n-1) + a(n-2)+((-1)^n)*A056565(n), n >= 2, a(0)=1, a(1)=55.
G.f.: exp( Sum_{k>=1} F(10*k)/F(k) * x^k/k ), where F(n) = A000045(n). - Seiichi Manyama, May 07 2025

A056568 Fibonomial coefficients.

Original entry on oeis.org

1, 89, 12816, 1493064, 187628376, 22890661872, 2824135408458, 346934172869802, 42689423937884208, 5249543573067466872, 645693859487298425256, 79413089729752455762384, 9767258556969762111163771, 1201288963378036364032704659, 147748983166877427393815516256

Offset: 0

Wolfdieter Lang, Jul 10 2000

Alois P. Heinz, Table of n, a(n) for n = 0..150

Cf. A010048, A000045, A001654-8, A056565-7, A001906, A004187 (signed), A049660, A049668 (signed), A049670.

[&*[Fibonacci(n+i): i in [0..9]]/122522400: n in [1..15]]; // Vincenzo Librandi, Oct 31 2014

F:= combinat[fibonacci]: a:= n-> mul(F(n+i), i=0..9)/122522400: seq(a(n), n=1..18); # Zerinvary Lajos, Oct 07 2007
a:= n-> (Matrix(11, (i,j)-> if (i=j-1) then 1 elif j=1 then [1514513, -582505, -83215, 4895, 89, -1][abs(i-11/2)+1/2] else 0 fi)^n)[1, 1]; seq(a(n), n=0..18);  # Alois P. Heinz, Aug 15 2008

Times@@@Partition[Fibonacci[Range[30]],10,1]/122522400 (* Harvey P. Dale, Jul 27 2019 *)

a(n)=prod(k=0,9,fibonacci(n+k))/122522400; \\ Joerg Arndt, Oct 31 2014

a(n) = A010048(n+10,10) =: Fibonomial(n+10,10).
G.f.: 1/p(11,n) with p(11,n) = 1-89*x -4895*x^2 +83215*x^3 +582505*x^4 -1514513*x^5 -1514513*x^6 +582505*x^7 +83215*x^8 -4895*x^9 -89*x^10 +x^11 = (1+x) *(1-3*x+x^2) *(1+7*x+x^2) *(1-18*x+x^2) *(1+47*x+x^2) *(1-123*x+x^2) (n=8 row polynomial of signed Fibonomial triangle A055870; see this entry for Knuth and Riordan references).
Recursion: a(n)=123*a(n-1)-a(n-2)+((-1)^n)*A056566(n), n >= 2, a(0)=1, a(1)=89.
G.f.: exp( Sum_{k>=1} F(11*k)/F(k) * x^k/k ), where F(n) = A000045(n). - Seiichi Manyama, May 07 2025

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A010048 Triangle of Fibonomial coefficients, read by rows.

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A055870 Signed Fibonomial triangle.

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A056566 Fibonomial coefficients.

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A056567 Fibonomial coefficients.

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A056568 Fibonomial coefficients.

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