A277202 Ratio of the fibonomial Catalan numbers and Lucas numbers.

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

Offset: 1

Author

Vladimir Reshetnikov, Oct 04 2016

Keywords

• nonn
• easy

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.

Links

• 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.

Crossrefs

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

Programs

Mathematica

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 *)

Formula

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.