Why do you want to do this without a loop? When the purpose is an acceleration, omit the expensive factorial and power operations at first:
function M = Iterations(N)
F = [1, cumprod(1:21)];
sF = F;
sF(2:2:22) = -sF(2:2:22);
FN = F(N+1);
M = zeros(N+1, N+1);
for i = 0:N/2
t = N - i + 1;
b = F(t) ./ F(i+1:t) ./ sF(t-i:-1:1) .* (FN ./ F(t) ./ F(i+1));
M(i+1, i+1:t) = b;
M(i+1:t, i+1) = b;
end
end
For N = 10 this is needs 16% of the runtime of the original.
But due to the factorial the output will be accurate for N <= 21. If you need such a small set of inputs only, it would be more efficient, to create a list of possible values once and store it in a persistent varibale.
function M = IterationsX(N)
persistent List
if isempty(List)
List = cell(1, 21);
for k = 1:21
List{k} = Iterations(k);
end
end
M = List{N};
end
