Polynomial sequence?

[Antti Karttunen]

Nick Hobson wrote:

> Hi,
>
> What do people think of this sequence: 1, 7, 58, 601, 7656, 116929,  
> 2092112, 43006401, 999637120, 25933795801, ... ?
>
> The formula is a(n) = (n+1)^n - n!.  Lest that seem completely 
> artificial,  I should comment on how I stumbled across the sequence!  
> Fit a polynomial  f of degree n-1 to the first n nth powers of 
> positive integers.  Then  f(n+1) = a(n).
>
> For example, the quadratic that fits (1,1), (2,8), and (3,27) is f(n) 
> =  6n^2-11n+6.  Then f(4) = a(3) = 58.  Of course, it's not necessary 
> to  actually determine the polynomial f; a(n) can be found by 
> considering  differences.
>
It's much less artificial than many sequences that are proposed here, 
and like
you explain, it has some actual use and relevance.
So, in my opinion, it deserves its place among the A-numbers!

> Nick
>


Cheers,

Antti