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A003150 Fibonomial Catalan numbers.
(Formerly M3077)
5
1, 1, 3, 20, 364, 17017, 2097018, 674740506, 568965009030, 1255571292290712, 7254987185250544104, 109744478168199574282739, 4346236474244131564253156182, 450625464087974723307205504432150 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

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.

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.

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.

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.

Eric Weisstein's World of Mathematics, q-Binomial Coefficient.

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)_{-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

EXAMPLE

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

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, 1, n}]; Table[a[n], {n, 0, 13}] (* Jean-Fran├žois 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 *)

CROSSREFS

Cf. A010048, A000045, A003267.

Sequence in context: A163138 A201824 A203519 * A326869 A203194 A322455

Adjacent sequences:  A003147 A003148 A003149 * A003151 A003152 A003153

KEYWORD

nonn,easy,nice

AUTHOR

N. J. A. Sloane, Henry Gould

STATUS

approved

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Last modified November 13 04:20 EST 2019. Contains 329085 sequences. (Running on oeis4.)