[seqfan] A sequence relating to A157019
franktaw at netscape.net
franktaw at netscape.net
Wed Dec 2 12:01:44 CET 2009
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
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