|
%I M4272 #70 Nov 12 2023 13:16:08
%S 1,1,6,60,1820,136136,27261234,14169550626,19344810307020,
%T 69056421075989160,645693859487298425256,15803204856220738696714416,
%U 1012673098498882654470985390406,169885799961166470686816475170920550,74614732877610423587753604318734054624100
%N Central Fibonomial coefficients.
%C 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
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A. Brousseau, <a href="http://www.fq.math.ca/Scanned/6-1/brousseau3.pdf">A sequence of power formulas</a>, Fib. Quart., 6 (1968), 81-83.
%H A. Brousseau, <a href="http://www.fq.math.ca/fibonacci-tables.html">Fibonacci and Related Number Theoretic Tables</a>, Fibonacci Association, San Jose, CA, 1972, p. 74.
%H Phakhinkon Phunphayap, <a href="http://ithesis-ir.su.ac.th/dspace/bitstream/123456789/3040/1/59305804.pdf">Various Problems Concerning Factorials, Binomial Coefficients, Fibonomial Coefficients, and Palindromes</a>, Ph. D. Thesis, Silpakorn University (Thailand 2021).
%H Phakhinkon Phunphayap, Prapanpong Pongsriiam, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL21/Pongsriiam/pong12.html">Explicit Formulas for the p-adic Valuations of Fibonomial Coefficients</a>, J. Int. Seq. 21 (2018), #18.3.1.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/CentralFibonomialCoefficient.html">Central Fibonomial Coefficient</a> [From _Eric W. Weisstein_, Dec 08 2009]
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/q-BinomialCoefficient.html">q-Binomial Coefficient</a>.
%F 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
%F 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
%F 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
%F 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
%F a(n) = A010048(2*n, n). - _Vladimir Reshetnikov_, Sep 27 2016
%p with(combinat): a := n -> product(fibonacci(n+k+1), k=0..n-1)/product(fibonacci(k), k=1..n):
%p seq(a(n), n=0..20);
%t 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 *)
%t 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_ *)
%t 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 *)
%o (PARI) a(n)=prod(k=0,n-1,fibonacci(n+k+1))/prod(k=1,n,fibonacci(k))
%o for(n=0,14,print1(a(n),","))
%Y Bisection of A003268. Cf. A008341.
%Y Cf. A001622, A062073, A062381.
%K nonn,easy
%O 0,3
%A _N. J. A. Sloane_
%E More terms from _Sascha Kurz_ and _Rick L. Shepherd_, Mar 24 2002
%E a(1) = 1 added by _N. J. A. Sloane_, Dec 06 2009
%E Typo in second formula corrected by _Vaclav Kotesovec_, Apr 10 2015
%E Offset corrected from 1 to 0, formulae and programs are updated accordingly by _Vladimir Reshetnikov_, Sep 27 2016
|
