Q: Lambda function with number of arguments determined at compile-time I would like to declare a lambda function with exactly N parameters, where N is a template argument. Something like...
template <int N>
class A {
    std::function<void (double, ..., double)> func;
                        // exactly n inputs
};

I could not think of a way to do this with the metafunction paradigm.

A: A meta template that takes a template, a count, and a type, and invokes the template with N copies of the type:
template<template<class...>class target, unsigned N, class T, class... Ts>
struct repeat_type_N: repeat_type_N<target, N-1, T, T, Ts...> {};
template<template<class...>class target, class T, class... Ts>
struct repeat_type_N<target, 0, T, Ts...> {
  typedef target<Ts...> type;
};
template<template<class...>class target, unsigned N, class T>
using repeat_type_N_times = typename repeat_type_N<target, N, T>::type;

Now, we use it:
template<typename... Ts> using operation=void(Ts...);
template<unsigned N, class T> using N_ary_op = repeat_type_N_times< operation, N, T >;
template<unsigned N> using N_double_func = N_ary_op<N,double>;

And we test it:
void three_doubles(double, double, double) {}

int main() {
  N_double_func<3>* ptr = three_doubles;
  std::function< N_double_func<3> > f = three_doubles;
}

and win.
What exactly you use the double, double, double for is completely up to you in the above system.  You can have a lambda that you initialize a std::function with, for example.
You can pack up the double, double, double into a template<class...>struct type_list{}; so you can pass it as one argument to another template, then specialize to unpack it. 
A repeat_type that has less recursion for large N:
// package for types.  The typedef saves characters later, and is a common pattern in my packages:
template<class...>struct types{typedef types type;};

// Takes a target and a `types`, and applies it.  Note that the base has no implementation
// which leads to errors if you pass a non-`types<>` as the second argument:
template<template<class...>class target, class types> struct apply_types;
template<template<class...>class target, class... Ts>
struct apply_types<target, types<Ts...>>{
  typedef target<Ts...> type;
};
// alias boilerplate:
template<template<class...>class target, class types>
using apply_types_t=typename apply_types<target,types>::type;

// divide and conquer, recursively:
template<unsigned N, class T, class Types=types<>> struct make_types:make_types<
  (N+1)/2, T, typename make_types<N/2, T, Types>::type
> {};

// terminate recursion at 0 and 1:
template<class T, class... Types> struct make_types<1, T, types<Types...>>:types<T,Types...> {};
template<class T, class Types> struct make_types<0, T, Types>:Types{};

// alias boilerplate:
template<unsigned N, class T>
using make_types_t=typename make_types<N,T>::type;

// all of the above reduces `repeat_type_N_t` to a one-liner:    
template<template<class...>class target, unsigned N, class T>
using repeat_type_N_times = apply_types_t<target, make_types_t<N,T>>;

For large N, the above can significantly reduce compile times, and deal with overflowing the template stack.

A: You can write a template n_ary_function with a nested typedef type. This type can be used as follows:
template <int N>
class A {
    typename n_ary_function<N, double>::type func;
};

The following code fragment contains the definition of n_ary_function:
template <std::size_t N, typename Type, typename ...Types>
struct n_ary_function
{
    using type = typename n_ary_function<N - 1, Type, Type, Types...>::type;
};

template <typename Type, typename ...Types>
struct n_ary_function<0, Type, Types...>
{
    using type = std::function<void(Types...)>;
};


A: You can't do this directly.
You can do something like this
template <unsigned N> class UniformTuple;

template <>
class UniformTuple <0>
{
};

template <unsigned N>
class UniformTuple : public UniformTuple <N-1>
{
public:

    template <typename... Args>
    UniformTuple (double arg, Args... args)
    : UniformTuple <N-1> (args...)
    , m_value (arg)
    {
    }

private:

    double m_value;
};

template <int N>
class A
{
    std :: function <void (const UniformTuple <N> &)> func;
};


A: For completeness, here is a solution without recursion:
template <class Ret, class Arg, class Idx>
struct n_ary_function_;

template <class Ret, class Arg, std::size_t... Idx>
struct n_ary_function_<Ret, Arg, std::index_sequence<Idx...>> {
    template <class T, std::size_t>
    using id = T;

    using type = std::function<Ret(id<Arg, Idx>...)>;
};

template <class Ret, class Arg, std::size_t N>
using n_ary_function = typename n_ary_function_<
    Ret, Arg, std::make_index_sequence<N>
>::type;

See it live on Coliru
