I wrote:
> Jon Hough's suggestion looks very promising:
>
> x^y = (1 +(x-1))^y
> e^pi = (1+(e-1))^pi = 1+ pi*e + pi*(pi - 1)*e*e/2! +...
> http://en.wikipedia.org/wiki/Binomial_series
Ok, here's a whack at that:
binser =: (( ( (#@:] # {.@:[) ; (-~{:)~ ; >:@:]) )&> <\@:i.)~ <@,
('x^k';('y-i.k');'k!') , 5 binser 7 4
+---------+---------+---------+
|x^k |y-i.k |k! |
+---------+---------+---------+
|7 |4 |1 |
+---------+---------+---------+
|7 7 |4 3 |1 2 |
+---------+---------+---------+
|7 7 7 |4 3 2 |1 2 3 |
+---------+---------+---------+
|7 7 7 7 |4 3 2 1 |1 2 3 4 |
+---------+---------+---------+
|7 7 7 7 7|4 3 2 1 0|1 2 3 4 5|
+---------+---------+---------+
1 + +/ *`%/"1 */&> 5 binser 7 4
4096
Note the 1 + ... part would not be required if <\ included the empty
prefix (which is really neat if you think about it). Unfortunately, while
this seems to work ok for 7^4, it breaks on 4^7, 7^4.5, and basically
everything else. But I think I've played with this enough for the moment.
-Dan
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