A003267 Central Fibonomial coefficients.

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

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

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

• 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.

Crossrefs

Bisection of A003268. Cf. A008341.

Cf. A001622, A062073, A062381.

Programs

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 *)

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),","))

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)A001622%20is%20the%20golden%20ratio,%20and%20C(n,%20k)_q%20is%20the%20q-binomial%20coefficient.%20-%20_Vladimir%20Reshetnikov">{-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

Extensions

More terms from Sascha Kurz and Rick L. Shepherd, Mar 24 2002

a(1) = 1 added by N. J. A. Sloane, Dec 06 2009

Typo in second formula corrected by Vaclav Kotesovec, Apr 10 2015

Offset corrected from 1 to 0, formulae and programs are updated accordingly by Vladimir Reshetnikov, Sep 27 2016