If you want a simple low pass digital filter you can use code like this:
float temp1;
void setup ()
{
temp1 = analogRead (A0) ;
}
void loop ()
{
int input = analogRead (A0) ;
temp1 += (input - temp1) * 0.1 ; // classic 1 pole low-pass filter in numerically stable form.
....
}
Much less state is needed, since its an IIR filter, not an FIR filter (google/wikipedia are your friends!)
Your FIR filter takes 10 samples before its output is equal to the input even if the input is constant,
this filter is correctly initialized.