A003267 -id:A003267 - 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

A062073 Decimal expansion of Fibonacci factorial constant.

Original entry on oeis.org

1, 2, 2, 6, 7, 4, 2, 0, 1, 0, 7, 2, 0, 3, 5, 3, 2, 4, 4, 4, 1, 7, 6, 3, 0, 2, 3, 0, 4, 5, 5, 3, 6, 1, 6, 5, 5, 8, 7, 1, 4, 0, 9, 6, 9, 0, 4, 4, 0, 2, 5, 0, 4, 1, 9, 6, 4, 3, 2, 9, 7, 3, 0, 1, 2, 1, 4, 0, 2, 2, 1, 3, 8, 3, 1, 5, 3, 1, 2, 1, 6, 8, 4, 5, 2, 6, 2, 1, 5, 6, 2, 4, 9, 4, 7, 9, 7, 7, 4, 1, 2, 5, 9, 1, 3

Offset: 1

Jason Earls, Jun 27 2001

The Fibonacci factorial constant is associated with the Fibonacci factorial A003266.

Two closely related constants are A194159 and A194160. [Johannes W. Meijer, Aug 21 2011]

			1.226742010720353244417630230455361655871409690440250419643297301214...

Steven R. Finch, Mathematical Constants, Cambridge, 2003, Section 1.2.5.
R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics, Addison Wesley, 1990, pp. 478, 571.

Harry J. Smith, Table of n, a(n) for n=1..5000
M. Griffiths, Symmetric rational expressions in the Fibonacci numbers, Fib. Q., 46/47 (2008/2009), 262-267. [_N. J. A. Sloane_, Dec 05 2009]
Simon Plouffe, Fibonacci factorials
Eric Weisstein's World of Mathematics, Fibonacci Factorial Constant

Cf. A003266, A003267, A003268, A056569, A062072, A062381, A135407, A181926.

Cf. A218490, A253924, A256831, A259314, A259405.

RealDigits[N[QPochhammer[-1/GoldenRatio^2], 105]][[1]] (* Alonso del Arte, Dec 20 2010 *)
RealDigits[N[Re[(-1)^(1/24) * GoldenRatio^(1/12) / 2^(1/3) * EllipticThetaPrime[1,0,-I/GoldenRatio]^(1/3)], 120]][[1]] (* Vaclav Kotesovec, Jul 19 2015, after Eric W. Weisstein *)

\p 1300 a=-1/(1/2+sqrt(5)/2)^2; prod(n=1,17000,(1-a^n))

{ default(realprecision, 5080); p=-1/(1/2 + sqrt(5)/2)^2; x=prodinf(k=1, 1-p^k); for (n=1, 5000, d=floor(x); x=(x-d)*10; write("b062073.txt", n, " ", d)) } \\ Harry J. Smith, Jul 31 2009

C = (1-a)*(1-a^2)*(1-a^3)... 1.2267420... where a = -1/phi^2 and where phi is the Golden ratio = 1/2 + sqrt(5)/2.
C = QPochhammer[ -1/GoldenRatio^2]. [Eric W. Weisstein, Dec 01 2009]
C = A194159 * A194160. [Johannes W. Meijer, Aug 21 2011]
C = exp( Sum_{k>=1} 1/(k*(1-(-(3+sqrt(5))/2)^k)) ). - Vaclav Kotesovec, Jun 08 2013
C = Sum_{k = -inf .. inf} (-1)^((k-1)*k/2) / phi^((3*k-1)*k), where phi = (1 + sqrt(5))/2. - Vladimir Reshetnikov, Sep 20 2016

A003150 Fibonomial Catalan numbers.

Original entry on oeis.org

1, 1, 3, 20, 364, 17017, 2097018, 674740506, 568965009030, 1255571292290712, 7254987185250544104, 109744478168199574282739, 4346236474244131564253156182, 450625464087974723307205504432150, 122319234225590858340579679211039433810

Offset: 0

N. J. A. Sloane, Henry Gould

			a(5) = F(10)...F(7)/(F(5)...F(1)) = 55*34*21*13/(5*3*2*1*1) = 17017.

H. W. Gould, Fibonomial Catalan numbers: arithmetic properties and a table of the first fifty numbers, Abstract 71T-A216, Notices Amer. Math. Soc, 1971, page 938.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n=0..40
Christian Ballot, Lucasnomial Fuss-Catalan Numbers and Related Divisibility Questions, J. Int. Seq., Vol. 21 (2018), Article 18.6.5.
Paul Barry, On Integer-Sequence-Based Constructions of Generalized Pascal Triangles, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.4.
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.
Henry W. Gould, A new primality criterion of Mann and Shanks and its relation to a theorem of Hermite with extension to Fibonomials, Fib. Quart., 10 (1972), 355-364, 372.
Henry W. Gould, Fibonomial Catalan numbers: arithmetic properties and a table of the first fifty numbers, Abstract 71T-A216, Notices Amer. Math. Soc, 1971, page 938. [Annotated scanned copy of abstract]
Henry W. Gould, Letter to N. J. A. Sloane, Nov 1973, and various attachments.
Bruce Sagan, Open Problems for Catalan Number Analogues, January 11, 2015. See FiboCatalan numbers p. 6.
Eric Weisstein's World of Mathematics, q-Binomial Coefficient.

Cf. A000045, A001622, A003267, A010048, A062073.

QBinomial:= func< n, k, q | (&*[( 1-q^(n-j) )/( 1-q^(j+1) ): j in [0..k-1]]) >;
A003150:= func< n | n eq 0 select 1 else Round( ((1+Sqrt(5))/2)^(n^2)*QBinomial( 2*n, n, -2/(3+Sqrt(5)) )/Fibonacci(n+1) ) >;
[A003150(n): n in [0..30]]; // G. C. Greubel, Nov 04 2022

A010048 := proc(n,k) local a,j ; a := 1 ; for j from 0 to k-1 do a := a*combinat[fibonacci](n-j)/combinat[fibonacci](k-j) ; end do: return a; end proc:
A003150 := proc(n) A010048(2*n,n)/combinat[fibonacci](n+1) ; end proc:
seq(A003150(n),n=0..20) ; # R. J. Mathar, Dec 06 2010

f[n_]:= f[n]= Fibonacci[n]; a[n_]:=Product[f[k], {k,n+2,2n}]/Product[f[k], {k,n}]; Table[a[n], {n, 0, 13}] (* Jean-François Alcover, Dec 14 2011 *)
Table[Fibonorial[2 n]/(Fibonorial[n] Fibonorial[n+1]), {n, 0, 20}] (* Since v. 10.0, Vladimir Reshetnikov, May 21 2016 *)
Round@Table[GoldenRatio^(n^2) QBinomial[2 n, n, -1/GoldenRatio^2]/Fibonacci[n + 1], {n, 0, 20}] (* Round is equivalent to FullSimplify here, but is much faster - Vladimir Reshetnikov, Sep 25 2016 *)

ft(n) = prod(k=1, n, fibonacci(k)); \\ A003266
fn(n,k) = ft(n)/(ft(k)*ft(n-k)); \\ A010048
a(n) = fn(2*n, n)/fibonacci(n+1); \\ Michel Marcus, Aug 05 2023

def A003150(n): return round( golden_ratio^(n^2)*gaussian_binomial(2*n, n, -1/golden_ratio^2)/fibonacci(n+1) )
[A003150(n) for n in range(30)] # G. C. Greubel, Nov 04 2022

F(2n)*F(2n-1)* ...* F(n+2)/(F(n)*F(n-1)* ... *F(1)) = A010048(2*n,n)/F(n+1), F = Fibonacci numbers.
a(n) ~ sqrt(5) * phi^(n^2-n-1) / C, where phi = A001622 = (1+sqrt(5))/2 is the golden ratio and C = A062073 = 1.22674201072035324441763... is the Fibonacci factorial constant. - Vaclav Kotesovec, Apr 10 2015
a(n) = A003267(n)/F(n+1) = A010048(2*n, n)/F(n+1) = phi^(n^2) * C(2*n, n)A001622%20is%20the%20golden%20ratio,%20and%20C(n,%20k)_q%20is%20the%20q-binomial%20coefficient.%20-%20_Vladimir%20Reshetnikov">{-1/phi^2} / F(n+1), where phi = (1+sqrt(5))/2 = A001622 is the golden ratio, and C(n, k)_q is the q-binomial coefficient. - _Vladimir Reshetnikov, Sep 27 2016

A271421 a(n) = fibonorial(3n)/(fibonorial(2n+1)*fibonorial(n+1)), where fibonorial(n) = A003266(n).

Original entry on oeis.org

1, 4, 119, 23496, 32149806, 300214157831, 19246160432331107, 8451529006578585976752, 25443734373070679510011112460, 524973397889459587964008354031908560, 74243674067972394056586805754940632245000310, 71965837912588688126721254257169744333502564695515911

Offset: 1

Vladimir Reshetnikov, May 21 2016

Steven R. Finch, Mathematical Constants, Cambridge, 2003, Section 1.2.5.

Simon Plouffe, Fibonacci factorials.
Eric Weisstein's World of Mathematics, Fibonorial, Fibonacci Factorial Constant.

Cf. A003266, A003267, A003268, A062073, A003150, A000045.

Table[Fibonorial[3 n]/(Fibonorial[2 n + 1] Fibonorial[n + 1]), {n, 1, 30}] (* The sequence itself *)
QPochhammer[-1/GoldenRatio^2] (* The Fibonacci factorial constant C in the asymptotic expansion *)

a(n) ~ 5*phi^(2*n^2 - 3*n - 2)/C where phi = (1+sqrt(5))/2, and C = (-1/phi^2; -1/phi^2)_inf is the Fibonacci factorial constant whose decimal expansion is given in A062073.

A277202 Ratio of the fibonomial Catalan numbers and Lucas numbers.

Original entry on oeis.org

1, 1, 5, 52, 1547, 116501, 23266914, 12105638490, 16520674898562, 58983635652443448, 551479789789947609461, 13497628802000408584637131, 864924115332005227077169874150, 145099921975789867545171624212383670

Offset: 1

Vladimir Reshetnikov, Oct 04 2016

H. W. Gould, Fibonomial Catalan numbers: arithmetic properties and a table of the first fifty numbers, Abstract 71T-A216, Notices Amer. Math. Soc, 1971, page 938.

H. W. Gould, A new primality criterion of Mann and Shanks and its relation to a theorem of Hermite with extension to Fibonomials, Fib. Quart., 10 (1972), 355-364, 372.
Eric Weisstein's World of Mathematics, q-Binomial Coefficient, Lucas Number.

Cf. A000032, A010048, A000045, A003267, A003150.

Table[Fibonorial[2 n]/(Fibonorial[n] Fibonorial[n + 1] LucasL[n]), {n, 1, 15}] (* since version 10.0, or *)
Round@Table[GoldenRatio^(n^2) QBinomial[2 n, n, -1/GoldenRatio^2]/(Fibonacci[n + 1] LucasL[n]), {n, 1, 15}] (* Round is equivalent to FullSimplify here, but is much faster *)

a(n) = A003150(n)/A000032(n).

a(n) ~ sqrt(5) * phi^(n^2-2*n-1) / C, where phi = A001622 = (1+sqrt(5))/2 is the golden ratio and C = A062073 = (-1/phi^2)_inf = 1.22674201072035324441763... is the Fibonacci factorial constant.

cp's OEIS Frontend

A092779 Exponent of 2 in central fibonomial coefficient A003267.

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A010048 Triangle of Fibonomial coefficients, read by rows.

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A062073 Decimal expansion of Fibonacci factorial constant.

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A003150 Fibonomial Catalan numbers.

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A271421 a(n) = fibonorial(3*n)/(fibonorial(2*n+1)*fibonorial(n+1)), where fibonorial(n) = A003266(n).

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A277202 Ratio of the fibonomial Catalan numbers and Lucas numbers.

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A295562 List of numbers whose middle Fibonomial coefficient (2n,n)_F is prime to 105.

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A271421 a(n) = fibonorial(3n)/(fibonorial(2n+1)*fibonorial(n+1)), where fibonorial(n) = A003266(n).