[seqfan] A sequence relating to A157019

[franktaw at netscape.net]

I was looking at this sequence, Sum_{d|n} binomial(n/d+d-2,d-1), which 
has generating function Sum_{n>=1} x^n/(1 - x^n)^n, and decided to look 
at the related g.f. Sum_{n>=1} x^n/(1 + x^n)^n.  This sequence starts:

1, 0, 2, -2, 2, 0, 2, -8, 8, 0, 2, -12, 2, 0, 32, -36, 2, 0, 2, -20, 
58, 0, 2, -136, 72, 0, 92, -28, 2, 0, 2, -272, 134, 0, 422, -288, 2, 0, 
184, -480, 2, 0, 2, -44, 1232, 0, 2, -2360, 926, 0, 308, -52, 2, 0, 
2004, -1176, 382, 0, 2, -4064, 2, 0, 6470, -5128

This appears to be Sum_{d|n} (-1)^(d+1) binomial(n/d+d-2,d-1); I 
haven't actually tried to prove it, but 100 terms match.

There are two very striking things about this sequence.  One is the 
0's, which appear to come for every n == 1 (mod 4).  The other is the 
2's, which appear to come for p-1, p an odd prime.

Franklin T. Adams-Watters