A003150 -id:A003150 - 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

A065097 a(n) = ((2n+1) + (2n-1) - 1)!/((2n+1)!*(2n-1)!).

Original entry on oeis.org

1, 1, 7, 66, 715, 8398, 104006, 1337220, 17678835, 238819350, 3282060210, 45741281820, 644952073662, 9183676536076, 131873975875180, 1907493251046152, 27767032438524099, 406472021074865382, 5979899192930226746, 88366931393503350700, 1311063521138246054410

Offset: 0

Len Smiley, Nov 11 2001

A Catalan-like formula using consecutive odd numbers. Recall that Catalan numbers (A000108) are given by ((n+1)+(n)-1)!/((n+1)!(n)!).
From David Callan, Jun 01 2006: (Start)
a(n) = number of Dyck (2n)-paths (i.e., semilength = 2n) all of whose interior returns to ground level (if any) occur at or before the (2n-2)-nd step, that is, they occur strictly before the midpoint of the path.
For example, a(2)=7 counts UUUUDDDD, UUUDUDDD, UUDUUDDD, UUDUDUDD, UUUDDUDD, UD.UUUDDD, UD.UUDUDD ("." denotes an interior return to ground level).
This result follows immediately from an involution on Dyck paths, due to Emeric Deutsch, defined by E->E, UPDQ -> UQDP (where E is the empty Dyck path; U=upstep, D=downstep and P,Q are arbitrary Dyck paths), because the involution is fixed-point-free on Dyck (2n)-paths and contains one path of the type being counted in each orbit.
a(n) = Sum_{k=0..n-1} C(2n-1-2k)*C(2k). This identity has the following combinatorial interpretation:
a(n) is the number of odd-GL-marked Dyck (2n-1)-paths. An odd-GL vertex is a vertex at location (2i,0) for some odd i >= 1 (path starts at origin). An odd-GL-marked Dyck path is a Dyck path with one of its odd-GL vertices marked. For example, a(2)=7 counts UUUDDD*, UUDUDD*, UD*UUDD, UDUUDD*, UD*UDUD, UDUDUD*, UUDDUD* (the * denotes the marked odd-GL vertex). (End)
a(n+1) = Sum_{k=0..n} C(k)*C(2*n+1-k), n >= 0, with C(n) = A000108(n), also gives the odd part of the bisection of the half-convolution of the Catalan sequence A000108 with itself. For the definition of the half-convolution of a sequence with itself see a comment on A201204. There one also finds the rule for the o.g.f. given below in the formula section. The even part of this bisection is found under A201205. - Wolfdieter Lang, Jan 05 2012
From Peter Bala, Dec 01 2015: (Start)
Let x = p/q be a positive rational in reduced form with p,q > 0. Define Cat(x) = (1/(2*p + q))*binomial(2*p + q, p). Then Cat(n) = Catalan(n). This sequence is Cat(n + 1/2) = (1/(4*n + 4))*binomial(4*n + 4, 2*n + 1). Cf. A265101 (Cat(n + 1/3)), A265102 (Cat(n + 1/4)) and A265103 (Cat(n + 1/5)).
Number of maximal faces of the rational associahedron Ass(2*n + 1, 2*n + 3). Number of lattice paths from (0, 0) to (2*n + 3, 2*n + 1) using steps of the form (1, 0) and (0, 1) and staying above the line y = (2*n + 1)/(2*n + 3)*x. See Armstrong et al. (End)
Also the number of ordered rooted trees with 2n nodes, most of which are leaves, i.e., the odd bisection of A358585. This follows from Callan's formula below. - Gus Wiseman, Nov 27 2022

			G.f.: 1 + x + 7*x^2 + 66*x^3 + 715*x^4 + 8398*x^5 + 104006*x^6 + ...

Alois P. Heinz, Table of n, a(n) for n = 0..834 (terms n = 1..100 from Harry J. Smith)
D. Armstrong, B. Rhoades, and N. Williams, Rational associahedra and noncrossing partitions arxiv:1305.7286 [math.CO], 2008.
Emeric Deutsch, An involution on Dyck paths and its consequences, Discrete Math., 204 (1999), no. 1-3, 163-166.

Cf. A003150 (for analog with consecutive Fibonacci numbers).
Equals (1/2) A048990. Cf. A000108, A001795, A201204, A201205, A265101, A265102, A265103.
Cf. A000891, A001263, A001622, A090181, A185650, A358585.

[Binomial(4*n-1, 2*n-1)/(2*n+1): n in [1..20]]; // Vincenzo Librandi, Dec 09 2015

seq(binomial(4*n-1,2*n-1)/(2*n+1), n=0..30); # Robert Israel, Dec 08 2015

a[ n_] := If[ n < 1, 0, Binomial[ 4 n - 1, 2 n - 1] / (2 n + 1)]; (* Michael Somos, Oct 25 2014 *)

combinat::dyckWords::count(2*n)/2 $ n = 1..26 // Zerinvary Lajos, Apr 25 2007

a(n) = { if(n==0, 1, (4*n - 1)!/((2*n + 1)!*(2*n - 1)!)) } \\ Harry J. Smith, Oct 07 2009

vector(20, n, binomial(4*n-1, 2*n-1)/(2*n+1)) \\ Altug Alkan, Dec 08 2015

A065097 = lambda n: hypergeometric([1-2*n,-2*n],[2],1)/2
[Integer(A065097(n).n(500)) for n in (1..20)] # Peter Luschny, Sep 22 2014

a(n) = binomial(4*n-1, 2*n-1)/(2*n+1).
a(n) = C(2n)/2 where C(n) is the Catalan number A000108. - David Callan, Jun 01 2006
G.f.: 1/2 + (sqrt(2)/2)/sqrt(1+sqrt(1-16*x)). - Vladeta Jovovic, Sep 26 2003
G.f.: 1 + 3F2([1, 5/4, 7/4], [2, 5/2], 16*x). - Olivier Gérard, Feb 16 2011
O.g.f.: (1 + (cata(sqrt(x)) + cata(-sqrt(x)))/2)/2, with the o.g.f. cata(x) of the Catalan numbers. See the W. Lang comment above. - Wolfdieter Lang, Jan 05 2012
a(n) = hypergeometric([1-2*n,-2*n],[2],1)/2. - Peter Luschny, Sep 22 2014
a(n) = A001448(n) / (4*n + 2) if n>0. - Michael Somos, Oct 25 2014
n*(2*n+1)*a(n) - 2*(4*n-1)*(4*n-3)*a(n-1) = 0. - R. J. Mathar, Oct 31 2015
O.g.f. is 1 + Revert( x*(1 + x)/(1 + 2*x)^4 ). - Peter Bala, Dec 01 2015
Sum_{n>=0} 1/a(n) = 39/25 + 4*Pi/(9*sqrt(3)) - 24*log(phi)/(25*sqrt(5)), where phi is the golden ratio (A001622). - Amiram Eldar, Mar 02 2023
From Peter Bala, Apr 29 2024: (Start)
For n >= 1, a(n) = (1/8)*Sum_{k = 0..2*n-1} (-1)^k * 4^(2*n-k)*binomial(2*n-1, k)*Catalan(k+1).
For n >= 1, a(n) = (1/8)*(16^n)*hypergeom([1 - 2*n, 3/2], [3], 1). (End)

a(0)=1 prepended by Alois P. Heinz, Nov 28 2021

A256799 Catalan number analogs for A099927, the generalized binomial coefficients for Pell numbers (A000129).

Original entry on oeis.org

1, 1, 6, 203, 40222, 46410442, 312163223724, 12237378320283699, 2796071362211148193590, 3723566980632561787914135870, 28901575272390972687956930234335380, 1307480498356321410289575304307661963042110, 344746842780849469098742541704318199701366091840620

Offset: 0

Tom Edgar, Apr 10 2015

nonn

One definition of the Catalan numbers is binomial(2*n,n) / (n+1); the current sequence models this definition using the generalized binomial coefficients arising from Pell numbers (A000129).

			a(5) = Pell(10)..Pell(7)/Pell(5)..Pell(1) = (2378*985*408*169)/(29*12*5*2*1) = 46410442.
a(3) = A099927(6,3)/Pell(3) = 2436/12 = 203.

Alois P. Heinz, Table of n, a(n) for n = 0..50
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.

Cf. A000129, A099927, A003150, A099929, A256831, A256832.

p:= n-> (<<2|1>, <1|0>>^n)[1, 2]:
a:= n-> mul(p(i), i=n+2..2*n)/mul(p(i), i=1..n):
seq(a(n), n=0..12);  # Alois P. Heinz, Apr 10 2015

Pell[m_]:=Expand[((1+Sqrt[2])^m-(1-Sqrt[2])^m)/(2*Sqrt[2])]; Table[Product[Pell[k],{k,1,2*n}]/(Product[Pell[k],{k,1,n}])^2 / Pell[n+1],{n,0,15}] (* Vaclav Kotesovec, Apr 10 2015 *)

P=[lucas_number1(n, 2, -1) for n in [0..30]]
[1/P[n+1]*prod(P[1:2*n+1])/(prod(P[1:n+1]))^2 for n in [0..14]]

a(n) = Pell(2n)Pell(2n-1)...Pell(n+2)/Pell(n)Pell(n-1)...Pell(1) = A099927(2*n,n)/Pell(n+1) = A099929(n)/Pell(n+1), where Pell(k) = A000129(k).

a(n) ~ 2^(3/2) * (1+sqrt(2))^(n^2-n-1) / c, where c = A256831 = 1.141982569667791206028... . - Vaclav Kotesovec, Apr 10 2015

A367252 a(n) is the number of ways to tile an n X n square as explained in comments.

Original entry on oeis.org

1, 0, 1, 4, 88, 3939, 534560, 185986304, 175655853776, 437789918351688, 2898697572048432368, 50698981110982431863735, 2342038257118692026082013568, 285250169294740386915765591840768, 91531011920509198679773321121428857296, 77312253225939431362091700178995800855209496

Offset: 0

Anna Tscharre, Nov 11 2023

nonn

Draw a Dyck path from (0,0) to (n,n) so the path always stays above the diagonal. Now section the square into horizontal rows of height one to the left of the path and tile these rows using 1 X 2 and 1 X 1 tiles. Similarly, section the part to the right of the path into columns with width one and tile these using 2 X 1 and 1 X 1 tiles. Furthermore, no 1 X 1 tiles are allowed in the bottom row.

Anna Tscharre, A possible 6 X 6 tiling

Special case of A003150.

Cf. A055010, A307082.

b:= proc(x, y) option remember; (F->
     `if`(x=0 and y=0, 1, `if`(x>0, b(x-1, y)*F(y-1), 0)+
     `if`(y>x, b(x, y-1)*F(x+1), 0)))(combinat[fibonacci])
    end:
a:= n-> b(n$2):
seq(a(n), n=0..15);  # Alois P. Heinz, Nov 11 2023

b[x_, y_] := b[x, y] = With[{F = Fibonacci},
     If[x == 0 && y == 0, 1,
     If[x > 0, b[x - 1, y]*F[y - 1], 0] +
     If[y > x, b[x, y - 1]*F[x + 1], 0]]];
a[n_] := b[n, n];
Table[a[n], {n, 0, 15}] (* Jean-François Alcover, Nov 14 2023, after Alois P. Heinz *)

a(n) == 1 (mod 2) <=> n in { A055010 }. - Alois P. Heinz, Nov 11 2023

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

A010048 Triangle of Fibonomial coefficients, read by rows.

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A065097 a(n) = ((2n+1) + (2n-1) - 1)!/((2n+1)!*(2n-1)!).

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A256799 Catalan number analogs for A099927, the generalized binomial coefficients for Pell numbers (A000129).

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A367252 a(n) is the number of ways to tile an n X n square as explained in comments.

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