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%I #6 Oct 05 2016 03:05:02
%S 1,1,5,52,1547,116501,23266914,12105638490,16520674898562,
%T 58983635652443448,551479789789947609461,13497628802000408584637131,
%U 864924115332005227077169874150,145099921975789867545171624212383670
%N Ratio of the fibonomial Catalan numbers and Lucas 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.
%H H. 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 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/q-BinomialCoefficient.html">q-Binomial Coefficient</a>, <a href="http://mathworld.wolfram.com/LucasNumber.html">Lucas Number</a>.
%F a(n) = A003150(n)/A000032(n).
%F 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.
%t Table[Fibonorial[2 n]/(Fibonorial[n] Fibonorial[n + 1] LucasL[n]), {n, 1, 15}] (* since version 10.0, or *)
%t 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 *)
%Y Cf. A000032, A010048, A000045, A003267, A003150.
%K nonn,easy
%O 1,3
%A _Vladimir Reshetnikov_, Oct 04 2016
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