Mathematicaの関数を拡張した
理論解析の参考にしてくれ

round[x_, t_: 1/2] :=
If[FractionalPart[x] < t, IntegerPart[x], IntegerPart[x] + 1]
SetAttributes[round, Listable]

integration[n_, t_: 1/2] := (bb = Table[b[i], {i, n}];
Integrate[
Boole[And @@
Append[Table[0 < bb[[i]] < 1, {i, n}],
Total[round[bb, t]] == round[Total[bb], t]]],
Sequence @@ Table[{bb[[i]], 0, 1}, {i, n}]])