Rational Binomial Coefficients

[Eddie W. Shore]

Introduction

Let p be a rational fraction, p = num/dem. The rational binomial coefficients of order n are defined by:

B_0(p) = 1

B_n(p) = COMB(p, n) = ( p * (p - 1) * (p - 2) * (p - 3) * ... * (p - n + 1) ) / n!

There are algorithms, but the program RATBIN uses the definition.

HP Prime Program RATBIN

Arguments: rational fraction, order

Code:


EXPORT RATBIN(p,n)
BEGIN
// 2018-12-26 EWS
// p-q, n
// Rational Binomial Coefficient
LOCAL X;
IF n==0 THEN
RETURN 1;
ELSE
IF n==1 THEN
RETURN p;
ELSE
RETURN QPI(ΠLIST(p-MAKELIST(X,X,0,n-1))/n!);
END;
END;
END;

* Note: the result is not always a fraction, but you can convert the answer to fraction by pressing [ a b/c ]

Blog Link: https://edspi31415.blogspot.com/2019/01/...ional.html

Examples:

b_2(1/2) = -1/8

b_3(1/2) = 1/16

b_4(1/2) = -5/128

b_5(1/2) = 7/256

Source:

Henrici, Peter. Computational Analysis With the HP-25 Calculator A Wiley-Interscience Publication. John Wiley & Sons: New York 1977 . ISBN 0-471-02938-6