You can use also Reduce[] function:
In[1]:= s18[n_] :=
1/a1 (a10 2^(18 n - 18) + a9 2^(16 n - 16) + a8 2^(14 n - 14) +
a7 2^(12 n - 12) + a6 2^(10 n - 10) + a5 2^(8 n - 8) +
a4 2^(6 n - 6) + a3 2^(4 n - 4) + a2 2^(2 n - 2))
In[2]:= Reduce[
s18[2] == 512 && s18[3] == 32276480 && s18[4] == 5967012429824 &&
s18[5] == 1434047386279215104 &&
s18[6] == 367859068909182204772352 &&
s18[7] == 95910600411254377903510519808 &&
s18[8] == 25108333074341105474794741072658432 &&
s18[9] == 6579768945390796136704466286370881536000 &&
s18[10] == 1724700842014699584005152525604862865902141440 &&
s18[11] ==
452110402686837008940790840661958866701897063989248, {a1, a2, a3,
a4, a5, a6, a7, a8, a9, a10}, Integers] /.
a1 -> 109395*70945875*638512875*2338875*1488375*49116375*30405375*\
425675250*10854718875
Out[2]= (a2 | a3 | a4 | a5 | a6 | a7 | a8 | a9 | a10) \[Element]
Integers &&
a2 == 178282935567271856830509942680720274438676605292968750000000000\
00000 &&
a3 == 136156643654118251436525836319588715317466888235467529296875000\
00000 &&
a4 == 111536831530413524163808099074400549387026854019371795654296875\
00000 &&
a5 == 809677853005611016306812369252803030577856150916344642639160156\
2500 && a6 ==
5010752484776427247876881679510552148398313965147862434387207031250 \
&& a7 ==
2535507653513230338348779077835354575248482195101289749145507812500 \
&& a8 ==
983483218762795551135714248560863732627156343337717056274414062500 &&
a9 == 25995043117205830436081336726502026767857373030219316482543945\
3125 && a10 ==
35117979774198190374919205068343776102717455854172706604003906250
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