A003267 - OEIS

A003267

Central Fibonomial coefficients.
(Formerly M4272)

1, 1, 6, 60, 1820, 136136, 27261234, 14169550626, 19344810307020, 69056421075989160, 645693859487298425256, 15803204856220738696714416, 1012673098498882654470985390406, 169885799961166470686816475170920550, 74614732877610423587753604318734054624100

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

COMMENTS

The largest prime factor of a(n): 1, 1, 3, 5, 13, 17, 89, 233, 233, 1597, 1597, 1597, 28657, 28657, 28657, 514229, 514229, 514229, 514229, 514229, 514229, 514229, 433494437, 433494437, 2971215073, ..., . The union of the above list is: 1, 3, 5, 13, 17, 89, 233, 1597, 28657, 514229, 433494437, 2971215073, 14736206161, 46165371073, 92180471494753, 99194853094755497, ... . - Robert G. Wilson v, Dec 04 2009

REFERENCES

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

LINKS

Table of n, a(n) for n=0..14.

A. Brousseau, A sequence of power formulas, Fib. Quart., 6 (1968), 81-83.

A. Brousseau, Fibonacci and Related Number Theoretic Tables, Fibonacci Association, San Jose, CA, 1972, p. 74.

Phakhinkon Phunphayap, Various Problems Concerning Factorials, Binomial Coefficients, Fibonomial Coefficients, and Palindromes, Ph. D. Thesis, Silpakorn University (Thailand 2021).

Phakhinkon Phunphayap, Prapanpong Pongsriiam, Explicit Formulas for the p-adic Valuations of Fibonomial Coefficients, J. Int. Seq. 21 (2018), #18.3.1.

Eric Weisstein's World of Mathematics, Central Fibonomial Coefficient [From Eric W. Weisstein, Dec 08 2009]

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

FORMULA

For n > 0, a(n) = (-1)^floor(n/2)*det(M(n, -1))/det(M(n, 0)) where M(n, m) is the n X n matrix with coefficient 1/F(i+j+m), i=1..n, j=1..n. - Benoit Cloitre, Jun 05 2004

For n > 0, a(n) = -(GoldenRatio^(n^2) QPochhammer[(-1)^n GoldenRatio^(-2 n), -GoldenRatio^-2, 1 + n])/((-1 + (-1)^n GoldenRatio^(-2 n)) QPochhammer[ -GoldenRatio^-2, -GoldenRatio^-2, n]). - Eric W. Weisstein, Dec 08 2009

a(n) ~ phi^(n^2) / 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) = phi^(n^2) * C(2*n, n)_{-1/phi^2}, where phi = (1+sqrt(5))/2 = A001622 is the golden ratio, and C(n, k)_q is the q-binomial coefficient. - Vladimir Reshetnikov, Sep 26 2016

a(n) = A010048(2*n, n). - Vladimir Reshetnikov, Sep 27 2016

MAPLE

with(combinat): a := n -> product(fibonacci(n+k+1), k=0..n-1)/product(fibonacci(k), k=1..n):

seq(a(n), n=0..20);

MATHEMATICA

f[n_] := Product[Fibonacci[n + k + 1]/Fibonacci[k + 1], {k, 0, n - 1}]; Array[f, 14, 0] (* Robert G. Wilson v, Dec 04 2009 *)

Flatten[{1, Table[Round[-(GoldenRatio^(n^2) QPochhammer[(-1)^n GoldenRatio^(-2 n), -GoldenRatio^-2, 1 + n])/((-1 + (-1)^n GoldenRatio^(-2 n)) QPochhammer[ -GoldenRatio^-2, -GoldenRatio^-2, n])], {n, 1, 15}]}] (* Vaclav Kotesovec, Apr 10 2015 after Eric W. Weisstein *)

Round@Table[GoldenRatio^(n^2) QBinomial[2 n, n, -1/GoldenRatio^2], {n, 0, 20}] (* Round is equivalent to FullSimplify here, but is much faster - Vladimir Reshetnikov, Sep 25 2016 *)

PROG

(PARI) a(n)=prod(k=0, n-1, fibonacci(n+k+1))/prod(k=1, n, fibonacci(k))

for(n=0, 14, print1(a(n), ", "))

CROSSREFS

Bisection of A003268. Cf. A008341.

Cf. A001622, A062073, A062381.

Sequence in context: A251184 A329319 A351165 * A271681 A010574 A363789

Adjacent sequences: A003264 A003265 A003266 * A003268 A003269 A003270

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane