cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A277202 Ratio of the fibonomial Catalan numbers and Lucas numbers.

Original entry on oeis.org

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

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Author

Vladimir Reshetnikov, Oct 04 2016

Keywords

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

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.

A346045 Decimal expansion of Sum_{k>=0} 1/(k! * A003266(k)).

Original entry on oeis.org

2, 5, 9, 0, 5, 6, 1, 4, 0, 6, 5, 6, 6, 1, 8, 0, 0, 8, 3, 7, 8, 7, 7, 2, 4, 2, 3, 0, 6, 1, 7, 2, 0, 2, 1, 3, 4, 6, 9, 1, 0, 1, 6, 8, 6, 3, 8, 7, 4, 0, 4, 5, 2, 9, 5, 6, 8, 2, 2, 5, 5, 6, 9, 8, 7, 0, 4, 0, 5, 2, 0, 7, 3, 0, 6, 9, 2, 4, 4, 9, 9, 0, 6, 9, 0, 6, 7, 8, 6, 5, 2, 2, 6, 5, 3, 0, 6, 1, 6, 5
Offset: 1

Views

Author

Amiram Eldar, Jul 02 2021

Keywords

Comments

This constant is irrational (Mingarelli, 2013).

Examples

			2.59056140656618008378772423061720213469101686387404...

Links

  • Angelo B. Mingarelli, Abstract factorials, Notes on Number Theory and Discrete Mathematics, Vol. 19, No. 4 (2013), pp. 43-76 (see p. 61); arXiv preprint, arXiv:0705.4299 [math.NT], 2007-2012.

Crossrefs

Cf. A003266, A062073.

Programs

  • Mathematica
    RealDigits[Sum[1/(n! * Fibonorial[n]), {n, 0, 50}], 10, 100][[1]]
  • PARI
    f(n) = prod(i=1, n, fibonacci(i)); \\ A003266
suminf(k=0, 1/(k!*f(k))) \\ Michel Marcus, Jul 05 2021

A372949 a(n) = 2*f(2*n)/(f(n)*f(n+2)) where f = A003266.

Original entry on oeis.org

1, 2, 8, 91, 2618, 199716, 39690618, 20689636692, 28215085220016, 100763710906257557, 942012688139052139766, 23056957423045790791793932, 1477460537993359748548214768630, 247860656992078740305125996374953260, 108861324945456389643061592667638024842480
Offset: 1

Views

Author

Kendra Killpatrick, May 17 2024

Keywords

Comments

Fibonacci analog of the super ballot numbers.
a(n) is also the generalized FiboCatalan number for r=1. Proof that the formula always gives a positive integer can be found in a recent paper of K. Killpatrick. The sequence is the Fibonacci analog of the super ballot numbers given by Gessel (A007054). The sequence is also the Fibonacci analog of the generalized Catalan numbers, J_r*(2n)!/(n!*(n+r+1)!) where J_r=(2r+1)!/r!, for r=1. Gessel defined the generalized Catalan numbers and proved they are integers.

Examples

			a(5) = 2*f(10)/(f(5)*f(7)) = 2*122522400/(30*3120) = 2618, where f=A003266.

Links

Crossrefs

Cf. A003266, A007054.

Programs

  • Mathematica
    a[n_]:=2Fibonorial[2n]/(Fibonorial[n]Fibonorial[n+2]); Array[a,15] (* Stefano Spezia, May 23 2024 *)

Formula

a(n) ~ 10 * phi^((n-3)*(n+1)) / A062073, where phi = A001622 is the golden ratio. - Vaclav Kotesovec, May 29 2024
Previous Showing 31-33 of 33 results.

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