**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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