[seqfan] need algebra proof

[quantum pkt]

Prover(s) will get full credit.
Thx in advance, Gary
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Conjecture:
Given polcoeff S(x), by way of example given A000041, the partition sequence = (1, 1, 2, 3, 5, 7, 11, 15, 22,...); then
S(x) = (1 + x + 2x^2 + 3x^3 + ...).
 
Then S(x) = some A(x) such that S(x) = A(x) / A(x^2).
Our conjecture states that A(x) = S(x) * S(x^2) * S(x^4) * S(x^8) * S(x^16) * ...).
 
Similarly, for S(x) = B(x) / B(x^3), our conjecture states that:
 
B(x) = S(x) * S(x^3) * S(x^9) * S(x^27) * S(x^81) * ...
 
...
Generally, for any k, in S(x) = C(x) / C(x^k).