be allowed 后用doing还是to do

be allowed to do sth.被允许干什么 是固定搭配
前面所说的allow sth. 是allow 加名词的用法，和这个是不同的。

be allowed doing
allow
[əˈlau]
vt.
允许, 许可(做某事)
I don't think he would allow it.
我想他不会允许的。
给予, 让…得到
The court allowed a sum of money to each child for clothing.
法院判给每个孩子一笔服装费。
承认, 认为
The referee refused to allow the goal.
裁判不承认这次进球。

现代英汉综合大辞典

allow
[əˈlau]
vt.
准许, 许可
容许, 听任, 允许, 让
给与, 让得到
承认(that)
酌留, 酌情增减(for)
想要, 打算
(由于不小心)让某种事情得以发生
allow a cake to burn
(由于不小心)让饼烤焦了
Smoking is not allowed here.
此处不准吸烟。
No dogs allowed.
禁止带狗。
A-me to introduce to you my friend Mr. Thomas.
请允许我把我的朋友托马斯先生介绍给你。
We don't allow such things to be done.
我们不容许有这种事情发生。
The manager allowed him one week of vacation in the summer.
夏天经理给他一星期的假期。
He is allowed ten dollars a month for his pocket money.
他