Polynomial sequence?

[franktaw at netscape.net]

My opinion: just the formula is pretty artificial; I wouldn't submit it 
based just on that.  But if you include the other characterization of 
the sequence, it becomes something which definitely does belong in the 
OEIS.

Franklin T. Adams-Watters


-----Original Message-----
From: nickh at qbyte.org

  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.

 Nick


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