Fibonomial Coefficient

Author

Eric W. Weisstein
December 8, 2009

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

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

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

Code

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

FibonomialCoefficient2[m_,n_]:=(LucasL[n]FibonomialCoefficient[m-1,n]+LucasL[m-n]FibonomialCoefficient[m-1,n-1])/2

Closed Form

V6

Product[Fibonacci[i],{i,k}]

Product[Fibonacci[i],{i,m-k+1,m}]

V7

Product[Fibonacci[i],{i,k}]

FullSimplify[Fibonacci[i]//FunctionExpand,i∈Integers&&i>0]

TimeConstrained[Product[((-(((-(2/(1+(Sqrt[5])))))^i)+((((1/2) (1+(Sqrt[5]))))^i))/(Sqrt[5])),{i,k}],60]//Timing

Product[Fibonacci[i],{i,m-k+1,m}]

TimeConstrained[Product[((-(((-(2/(1+(Sqrt[5])))))^i)+((((1/2) (1+(Sqrt[5]))))^i))/(Sqrt[5])),{i,m-k+1,m}],60]//Timing

Triangle

Table[FibonomialCoefficient2[n,k],{n,0,5},{k,0,n}]//Expand//ColumnForm

(tri=Table[FibonomialCoefficient[n,k],{n,0,5},{k,0,n}])//ColumnForm

Flatten[tri]