Seqfans, 
      David probably already noticed this, but I make some quick observations
by mere inspection (I am on my lunch break and lack time).   
      The reciprocal of the g.f.s of p_k seems to reveal what is going on 
with these nice sequences:
1/GF(p_2) = GF([1,-1,0,0,1,0,0,0,0,-1,0,0,0,0])
1/GF(p_3) = GF([1,-1,-1,0,0,1,0,1,0,0,0,0,-1,0])
1/GF(p_4) = GF([1,-1,-1,-1,0,0,1,0,1,0,1,0,0,0])
1/GF(p_5) = GF([1,-1,-1,-1,-1,0,0,1,0,1,0,1,0,1])
1/GF(p_6) = GF([1,-1,-1,-1,-1,-1,0,0,1,0,1,0,1,0])
1/GF(p_7) = GF([1,-1,-1,-1,-1,-1,-1,0,0,1,0,1,0,1])
1/GF(p_8) = GF([1,-1,-1,-1,-1,-1,-1,-1,0,0,1,0,1,0])
1/GF(p_9) = GF([1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,1,0,1])
1/GF(p_10) = GF([1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,1,0])

It appears that the g.f. of p_2 is 
     Sum_{n>=0}  (-1)^n*x^(n^2)
 
The g.f.s of the other p_k seem forthcoming.
      Paul
 

-- "David Wilson" <davidwwilson@comcast.net> wrote:
[...]
Computing a few terms of p_k for small k gives
p_1 = (1 0 0 0 0 0 0 0 0 0 0 0 0 0 ...)
p_2 = (1 1 1 1 0 -1 -2 -3 -3 -1 2 6 10 11 ...)
p_3 = (1 1 2 3 5 7 11 15 22 30 42 56 77 101 ...)
p_4 = (1 1 2 4 7 13 23 42 75 135 242 434 779 1396 ...)
p_5 = (1 1 2 4 8 15 29 55 106 202 387 739 1414 2702 ...)
p_6 = (1 1 2 4 8 16 31 61 119 234 458 898 1759 3447 ...)
p_7 = (1 1 2 4 8 16 32 63 125 247 490 970 1922 3806 ...)
p_8 = (1 1 2 4 8 16 32 64 127 253 503 1002 1994 3970 ...)
p_9 = (1 1 2 4 8 16 32 64 128 255 509 1015 2026 4042 ...)
p_10 = (1 1 2 4 8 16 32 64 128 256 511 1021 2039 4074 ...)
[...]
