call bitcoin API from Remote PC I want to make some test on testnet using the API and I would like to fetch datas from a remote computer. 
The client is on raspberry which has an internal IP,than I have a dyndns address which pings my router on a given port(3400) -> from router I redirect the request to raspberry PI's IP with clients port for API calls(192.168.178.40:8332) and I have this in my 

rpcallowip=192.168.178.40


but I get 403 status code
It isnt clear what you're doing, and I get the impression that you dont speak english natively, so I doubt we're going to get a better explanation of what you're trying to do. Given your question I can only assume your RaspberryPi's IP is 192.168.178.40, it is running bitcoind, and your trying to access the API on it from a computer somewhere on the internet (because you mention dyn-dns), and that there is a router in front of the RaspberryPi performing NAT. If you want any computer on the internet to be able to access the API

rpcallowip=0.0.0.0/0


or if you only want a single computer on the internet with a static IP to be able to access it, then you'd put in the IP address of that computer rather than the RaspberryPi's IP address.

If I got it wrong, try rephrasing your question, because as I said, we don't know how your network is configured.