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

1,3

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

Table of n, a(n) for n=1..14.

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.

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.

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 *)

Crossrefs

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

Sequence in context: A216464 A279644 A001173 * A223248 A208303 A215902

Adjacent sequences: A277199 A277200 A277201 * A277203 A277204 A277205

Keyword

nonn,easy

Author

Vladimir Reshetnikov, Oct 04 2016

Status

approved