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A nonlinear system

I am trying to solve a system of equations using this code:

bt=Table[Sum[a[k,i], {i,0,n-k}], {k,0,n}]
btt=Table[If[k==2,1,0], {k,0,n}]
ct=Table[(Sum[a[i,k], {i,0,n-k}])^2, {k,2,n}]
ctt=Table[If[k==2,1,0], {k,2,n}]
f2=Sum[a[k,i] x^k x^i, {k,0,n}, {i,0,n-k}]
f3=Sum[a[k,i] f2^k x^i, {k,0,n}, {i,0,n-k}]/x
f4=Sum[a[k,i] f3^k x^i, {k,0,n}, {i,0,n-k}]/f2
p1=f2 f4 x^n+x^(n^3+15)
g4=Sum[a[k,i] f2^k f2^i, {k,0,n}, {i,0,n-k}]
p2=f2 g4 x^n+x^(n^3+15)
eqns={bt==btt, ct==ctt, Sum[a[i,0], {i,0,n}]==0,

It is a conjecture that for any n>3 the solutions of this system are
1.  a[0,2]=1,  a[2,0]=1, a[1,1]=c, a[0,1]=-c,  a[1,0]=-c,  a[0,0]=c-1
2.  a[0,2]=-1, a[2,0]=1, a[0,1]=2, a[0,0]=-1
3.  a[2,2]=1
with arbitrary c, and all other a[i,j] being equal to zero.
Could anybody verify this for  some value of n>5 (if possible for n=10)
by using the above code?
For n=4, Mathematica 2.2 on my PC (40 MB ram) gives Out of memory.


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