Nesting n-sets of for loops - (Julia) - loops

Question

I am currently trying to create n-nested loops in order to compute a rather tricky sum.
Each term in the sum depends on the last, and therefore the nesting is required. Here is some Julia-written pseudo-code explaining what I'm trying to get:
# u,v are fixed integers
# m is a fixed, n-length array of integers
_
for i_1 in 0:min(m[1], u) |#Trio of Loops we want to repeat n times
for j_1 in 0:min(m[1], v) |
for t_1 in i_1:(i_1 + j_1) _|#1
_
for i_2 in 0:min(m[2], u - i_1) |
for j_2 in 0:min(m[2], v - j_1) |
for t_2 in i_2:(i_2 + j_2) _|#2
#... repeat until nth time...#
_
for i_n in 0:min(m[n],u - sum(#all i_'s up to n-1)) |
for j_n in 0:min(m[n],v - sum(#all j_'s up to n-1)) |
for t_n in i_n:(i_n + j_n) _|#n
#Sum over sum function which depends on i_'s and j_'s and t_'s
X += f(i_1,j_1,t_1)*f(i_2,j_2,t_2)*...*f(i_n,j_n,t_n)
I am completely unsure of how to properly code this for an arbitrary n-number of trios. If the number n is small then obviously this can be explicitly written, but if not, then we are stuck. I assume that there is a recursive way to employ this code, but I am unsure how to go about this.
Any and all help would be appreciated.

Answer 1

Here is a quick attempt at a slightly simplified version of your problem.
function inner(X, f, ms)
is = [Symbol(:i_, n) for n in 1:length(ms)]
js = [Symbol(:j_, n) for n in 1:length(ms)]
fs = [:($f($i, $j)) for (i,j) in zip(is, js)]
:($X += *($(fs...)))
end
inner(:X, :g, [3,4,5]) # :(X += g(i_1, j_1) * g(i_2, j_2) * g(i_3, j_3))
function looper(ex::Expr, ms, u)
isempty(ms) && return ex
n = length(ms)
m_n = ms[n]
i_n = Symbol(:i_,n)
j_n = Symbol(:j_,n)
out = :( for $i_n in 0:min($m_n, $u)
for $j_n in 0:min($m_n, $u)
$ex
end
end)
looper(out, ms[1:end-1], u)
end
looper(inner(:g, [4,5]), [4,5], 3)
# :(for i_1 = 0:min(4, 3)
# for j_1 = 0:min(4, 3)
# for i_2 = 0:min(5, 3)
# for j_2 = 0:min(5, 3)
# X += g(i_1, j_1) * g(i_2, j_2) ...
macro multi(X, f, ms, u)
ms isa Expr && ms.head == :vect || error("expected a literal vector like [1,2,3]")
looper(inner(esc(X), esc(f), ms.args), ms.args, esc(u))
end
#macroexpand #multi X f [4,5] u
# :(for var"#74#i_1" = 0:Main.min(4, u) ...
function multi(X, f, u)
#multi X f [4,5] u
X
end
multi(0.0, atan, 1)