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# HP Prime and Casio fx-5800p: Rational Binomial Coefficients

HP Prime and Casio fx-5800p:  Rational Binomial Coefficients

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

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 ]

Casio fx-5800p Program RATBIN

For fractional results, use the fraction button [  []/[] ].

“2018-12-26 EWS”
“FRACTION”? → P
“ORDER?” → N
If N=0
Then
0
IfEnd
If N=1
Then
1
IfEnd
If N>1
Then
Prod (P-Seq(X,X,0,N-1,1)) ÷ N! → Q
Q
IfEnd

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

Eddie

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