Encouraged and emboldened (embiggened?) by the ingenuity displayed by some of our readers, I’d like to see what sort of response we get to this Problem of the Week:
Establish the following identity:
for all natural numbers
.
(Here we make the convention 
Please send your solutions to topological[dot]musings[At]gmail[dot]com by Wednesday, June 11, 11:59pm (UTC); do not submit solutions in Comments. Everyone with a correct solution gets entered in our Hall of Fame! We look forward to your response.
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June 10, 2008 at 8:15 am
Joe
Hmm, that’s the second time I’ve seen the word embiggened used in the past five minutes.
http://unapologetic.wordpress.com/2008/05/30/blath-captcha/
June 10, 2008 at 11:39 am
Todd Trimble
Yeah, it’s from an episode of The Simpsons, where the legendary founder of Springfield, Jebediah Springfield, is quoted as having said, “A noble spirit embiggens the smallest man.” I acknowledge that John’s recent post may have played a role here, although it’s not self-evident to me because the word immediately came to mind when I wrote “emboldened”. Anyway…
June 10, 2008 at 5:07 pm
Paul Shearer
I’m not getting anywhere with generating functions or the bijections suggested by the formula. I hope somebody comes up with a good bijective proof, the formula is just begging for it!
June 10, 2008 at 8:07 pm
Todd Trimble
It is indeed! Watch this space though…
June 12, 2008 at 11:09 am
Solution to POW-4: A curious identity involving self-powers « Todd and Vishal’s blog
[...] Week (POW), Puzzles by Todd Trimble Tags: combinatorics, trees The solutions are in! I thought last week’s problem might have been a little more challenging than problems of previous weeks — the identity is [...]
June 12, 2008 at 3:42 pm
John Armstrong
Why are you so surprised? It’s a perfectly cromulent word.