You forgot to insert a conjunction between S18[10] and S18[11]. Replace comma with &&. You may also exclude a1 from the list of unknowns.
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));
NSolve[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, {a2, a3, a4, a5, a6, a7, a8, a9, a10}, Integers]
