DB48X: HP48-like RPL implementation for DM42

[Thomas Klemm]

(Yesterday 05:05 PM)n1msr Wrote:  It becomes important when the Comb[inations] function is used (…)

I just cobbled together this program for the HP-15C to calculate:

\(
\binom{n}{k} = \frac{n!}{k!(n-k)!}
\)

Code:

   001 {    44  0 } STO 0
   002 {    42  0 } f x!
   003 {       34 } X<=>Y
   004 {    42  0 } f x!
   005 {    43 36 } g LSTx
   006 { 45 30  0 } RCL - 0
   007 {    42  0 } f x!
   008 {       10 } /
   009 {       34 } X<=>Y
   010 {       10 } /

Example

5 ENTER 3
R/S

10.0000

Nothing to see here. That's what we expected.
But it also allows us to calculate values of \(\binom{0.5}{k}\) for consecutive \(k\).

Examples

0.5 ENTER 0
R/S

1.0000

0.5 ENTER 1
R/S

0.5000

0.5 ENTER 2
R/S

-0.1250

0.5 ENTER 3
R/S

0.0625

0.5 ENTER 4
R/S

-0.039062500

0.5 ENTER 5
R/S

0.027343750

0.5 ENTER 6
R/S

-0.020507813

Compare this to the Taylor series of:

\(
\begin{align}
\sqrt{1+x}
&= (1+x)^{\frac{1}{2}} \\
&= 1+\frac{x}{2}-\frac{x^2}{8}+\frac{x^3}{16}-\frac{5x^4}{128}+\frac{7x^5}{256}-\frac{21x^6}{1024}+\cdots
\end{align}
\)

As we can see in this example, there are good reasons to allow non-integer values in binomial coefficients.

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Hint: Don't try this with an HP 20b.