def countNecklaces(n, m):
N = n*m
R = PolynomialRing(QQ, m, 't')
ts = R.gens()
f = sum(t for t in ts)
R2 = PolynomialRing(R, N, 'a')
aGens = R2.gens()
ds = divisors(N)
F = 1/N * sum(euler_phi(N/d) * aGens[N/d-1]**d for d in ds)
g = F(*[f(*[t**k for t in ts]) for k in range(1, N+1)])
#print (g)
wantMono = prod(t**n for t in ts)
return g.coefficient(wantMono)
sol = countNecklaces(20, 3)
print (sol)
{"html5":"htmlmixed","css":"css","javascript":"javascript","php":"php","python":"python","ruby":"ruby","lua":"text\/x-lua","bash":"text\/x-sh","go":"go","c":"text\/x-csrc","cpp":"text\/x-c++src","diff":"diff","latex":"stex","sql":"sql","xml":"xml","apl":"apl","asterisk":"asterisk","c_loadrunner":"text\/x-csrc","c_mac":"text\/x-csrc","coffeescript":"text\/x-coffeescript","csharp":"text\/x-csharp","d":"d","ecmascript":"javascript","erlang":"erlang","groovy":"text\/x-groovy","haskell":"text\/x-haskell","haxe":"text\/x-haxe","html4strict":"htmlmixed","java":"text\/x-java","java5":"text\/x-java","jquery":"javascript","mirc":"mirc","mysql":"sql","ocaml":"text\/x-ocaml","pascal":"text\/x-pascal","perl":"perl","perl6":"perl","plsql":"sql","properties":"text\/x-properties","q":"text\/x-q","scala":"scala","scheme":"text\/x-scheme","tcl":"text\/x-tcl","vb":"text\/x-vb","verilog":"text\/x-verilog","yaml":"text\/x-yaml","z80":"text\/x-z80"}
