Central Fibonomial Coefficient

Author

Eric W. Weisstein
December 9, 2009

This notebook downloaded from http://mathworld.wolfram.com/notebooks/Combinatorics/CentralFibonomialCoefficient.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/CentralFibonomialCoefficient.html.

©2009 Wolfram Research, Inc. except for portions noted otherwise

Code

FibonomialCoefficient[m_,k_]:=Divide@@Times@@@Fibonacci/@{Range[m-k+1,m],Range[k]}
CentralFibonomialCoefficient[n_]:=FibonomialCoefficient[2n,n]

Enumeration

Table[CentralFibonomialCoefficient[n],{n,10}]

Table[Product[Fibonacci[n+k]/Fibonacci[k],{k,n}],{n,10}]

Closed Form

FindSequenceFunction

FindSequenceFunction[Table[CentralFibonomialCoefficient[n],{n,10}]]

FindSequenceFunction[Table[CentralFibonomialCoefficient[n],{n,20}]]

FindSequenceFunction[Table[CentralFibonomialCoefficient[n],{n,2,20}]]

Product

Product[Fibonacci[n+k]/Fibonacci[k],{k,n}]//Timing

expr=FullSimplify[-((((-1))^(-n)) ((((((-2))^n)))^n) (((-(((((2 I)/(1+(Sqrt[5])))))^(-2 n))))^n) (((1+(Sqrt[5])))^(-(n^2))) QPochhammer[-(((((2 I)/(1+(Sqrt[5])))))^n),((2 I)/(1+(Sqrt[5]))),1+n] QPochhammer[(((((2 I)/(1+(Sqrt[5])))))^n),((2 I)/(1+(Sqrt[5]))),1+n])/((-1+(((((2 I)/(1+(Sqrt[5])))))^(2 n))) QPochhammer[-(2/(3+(Sqrt[5]))),-(2/(3+(Sqrt[5]))),n]),n∈Integers&&n>0]

N[expr/.n->5]

expr2=expr/.{(1+(Sqrt[5]))->2GoldenRatio,(-3+(Sqrt[5]))->-2/GoldenRatio^2}

N[expr2/.n->5]

FullSimplify[expr2,n∈Integers&&n>0]

TraditionalForm[%]

l1=Table[-((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,10}]//FullSimplify

l2=FunctionExpand[%]//FullSimplify

l3=FunctionExpand[%]//Expand//FullSimplify

ByteCount/@{l1,l2,l3}

N[Table[-((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,10}],100]//Round