A003150 Fibonomial Catalan numbers.

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

Offset: 0

Author

N. J. A. Sloane, Henry Gould

Keywords

• nonn
• easy
• nice

Examples

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

References

• 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).

Links

• 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.

Crossrefs

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

Programs

Magma

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

Maple

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

Mathematica

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 *)

PARI

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

SageMath

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

Formula

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