Roman Coefficient

The Roman coefficient is a generalization of the binomial coefficient whose notation was suggested by Knuth,

 |_n; k]=(|_n]!)/(|_k]!|_n-k]!),

where |_n] is a Roman factorial. The above expression is read "Roman n choose k." Whenever the binomial coefficient is defined (i.e., n>=k>=0 or k>=0>n), the Roman coefficient agrees with it. However, the Roman coefficients are defined for values for which the binomial coefficients are not, e.g.,

|_n; -1]=1/(|_n+1])
|_0; k]=((-1)^(k+(k>0)))/(|_k]),


 n<0={1   for n<0; 0   for n>=0.

The Roman coefficients also satisfy properties like those of the binomial coefficient,

 |_n; k]=|_n; n-k]
 |_n; k]|_k; r]=|_n; r]|_n-r; k-r],

an analog of Pascal's formula

 |_n; k]=|_n-1; k]+|_n-1; k-1],

and a curious rotation/reflection law due to Knuth

 (-1)^(k+(k>0))|_-n; k-1]=(-1)^(n+(n>0))|_-k; n-1]

(Roman 1992).

See also

Binomial Coefficient, Roman Factorial

Explore with Wolfram|Alpha


Loeb, D. E. "A Generalization of the Binomial Coefficients." 9 Feb 1995., S. "The Logarithmic Binomial Formula." Amer. Math. Monthly 99, 641-648, 1992.

Referenced on Wolfram|Alpha

Roman Coefficient

Cite this as:

Weisstein, Eric W. "Roman Coefficient." From MathWorld--A Wolfram Web Resource.

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