[A000179]

[James Propp]

>It doesn't seem right that a combinatorial sequence should include 
>a negative number.

I agree, it does seem strange!

On a purely algebraic level, we might try to find the "right" way to
extend A000179 so that the sequence, viewed as a function, has all of
Z as its domain.  If we assume that the relation
	(n-2)*a(n) = (n^2-2*n)*a(n-1) + n*a(n-2) - 4*(-1)^n
holds for all integers n, then the facts that a(3) = 1 and a(2) = 0 force us 
to conclude that a(1) = -1, a(0) = 2, and a(-1) = -1.  Working farther back, 
we find that the value of a(-2) cannot be deduced, but that a(-3), a(-4), 
etc. can all recovered from a(-2).  If we put x = a(-2), we get

a(-2)=x
a(-3)=3*x+1
a(-4)=14*x+2
a(-5)=75*x+13
a(-6)=471*x+80
a(-7)=3402*x+579
etc.

Note that the choice x=0 gives us the original sequence A000179 in reverse.
So one natural way to extend the definition of a(n) is to put a(-n)=a(n)
for all n<0.  I have no idea what sort of combinatorial interpretation would
be appropriate, though, or what might constitute a combinatorial "story" to
explain the functional equation.

Jim Propp