Lua卡塔兰数：合法括号序列的计数

local function catalan(n)
    local function fact(k)
        local r = 1
        for i = 2, k do r = r * i end
        return r
    end
    return fact(2 * n) / (fact(n) * fact(n + 1))   -- 阶乘易溢出
end

local function catalan(n)
    local c = 1
    for i = 1, n do
        c = c * 2 * (2 * i - 1) / (i + 1)
    end
    return math.floor(c + 0.5)
end
sendmsg(actor, 1, "3对括号合法序列 "
    .. catalan(3) .. " 种，5对 " .. catalan(5) .. " 种")