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%I M3077 #72 Nov 12 2023 13:14:25
%S 1,1,3,20,364,17017,2097018,674740506,568965009030,1255571292290712,
%T 7254987185250544104,109744478168199574282739,
%U 4346236474244131564253156182,450625464087974723307205504432150,122319234225590858340579679211039433810
%N Fibonomial Catalan numbers.
%D 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.
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H T. D. Noe, <a href="/A003150/b003150.txt">Table of n, a(n) for n=0..40</a>
%H Christian Ballot, <a href="https://www.emis.de/journals/JIS/VOL21/Ballot/ballot30.html">Lucasnomial Fuss-Catalan Numbers and Related Divisibility Questions</a>, J. Int. Seq., Vol. 21 (2018), Article 18.6.5.
%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 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 Henry W. Gould, <a href="http://www.fq.math.ca/Scanned/10-4/gould1-a.pdf">A new primality criterion of Mann and Shanks and its relation to a theorem of Hermite with extension to Fibonomials</a>, Fib. Quart., 10 (1972), 355-364, 372.
%H Henry W. Gould, <a href="/A003099/a003099_2.pdf">Fibonomial Catalan numbers: arithmetic properties and a table of the first fifty numbers</a>, Abstract 71T-A216, Notices Amer. Math. Soc, 1971, page 938. [Annotated scanned copy of abstract]
%H Henry W. Gould, <a href="/A003099/a003099_1.pdf">Letter to N. J. A. Sloane, Nov 1973, and various attachments</a>.
%H Bruce Sagan, <a href="https://users.math.msu.edu/users/bsagan/Slides/OpcH.pdf">Open Problems for Catalan Number Analogues</a>, January 11, 2015. See FiboCatalan numbers p. 6.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/q-BinomialCoefficient.html">q-Binomial Coefficient</a>.
%F 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.
%F 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
%F a(n) = A003267(n)/F(n+1) = A010048(2*n, n)/F(n+1) = phi^(n^2) * C(2*n, n)_{-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
%e a(5) = F(10)...F(7)/(F(5)...F(1)) = 55*34*21*13/(5*3*2*1*1) = 17017.
%p 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:
%p A003150 := proc(n) A010048(2*n,n)/combinat[fibonacci](n+1) ; end proc:
%p seq(A003150(n),n=0..20) ; # _R. J. Mathar_, Dec 06 2010
%t 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 *)
%t Table[Fibonorial[2 n]/(Fibonorial[n] Fibonorial[n+1]), {n, 0, 20}] (* Since v. 10.0, _Vladimir Reshetnikov_, May 21 2016 *)
%t 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 *)
%o (Magma)
%o QBinomial:= func< n, k, q | (&*[( 1-q^(n-j) )/( 1-q^(j+1) ): j in [0..k-1]]) >;
%o 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) ) >;
%o [A003150(n): n in [0..30]]; // _G. C. Greubel_, Nov 04 2022
%o (SageMath)
%o def A003150(n): return round( golden_ratio^(n^2)*gaussian_binomial(2*n, n, -1/golden_ratio^2)/fibonacci(n+1) )
%o [A003150(n) for n in range(30)] # _G. C. Greubel_, Nov 04 2022
%o (PARI) ft(n) = prod(k=1, n, fibonacci(k)); \\ A003266
%o fn(n,k) = ft(n)/(ft(k)*ft(n-k)); \\ A010048
%o a(n) = fn(2*n, n)/fibonacci(n+1); \\ _Michel Marcus_, Aug 05 2023
%Y Cf. A000045, A001622, A003267, A010048, A062073.
%K nonn,easy,nice
%O 0,3
%A _N. J. A. Sloane_, _Henry Gould_
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