1. First, when do children get father's inheritance ?
In California, if father dies and left spouse and children, the spouse would take all of community property, and 1/3 to 1/2 of the father's separate property (depend on how many children, grandchildren), and the children divides up the remaining 1/2 to 2/3 of the father's separate property.
If father dies and there is no remaining spouse, the children would inherit in all father's property.
2. All father's Children have equal right. Father's children including all his children born in wedlock legally assumed to be his children, and all his out of wedlock children.
3. Out of wedlock children
Out of wedlock children means a child born out of wedlock. Some children might be born out of wedlock but conceived before a divorce, some children were born before mother and father got married, some children where born to mother who never ever married to the father. These are all out of wedlock children.
The out of wedlock children would have the EQUAL right like the marital children if the out-of-wedlock child was known to the father before he died, and he acknoweldge this child as his own in some way. Those children born to a mother who at some time was married to the father would most likely belong in this cateogory.
Things get hairy if the child was not known to the father, or the fauther knew but did not acknowledge this child as his own (e.g. you said he's mine, but I think Dave is his father, not me!). Because the father was dead before this child came forward, basically they need a paternity test. How the case turn out turns on many details and case law, and I really don't know too much about.
I suspect you were asking the last category of the out-of-wedlock, and sorry I can't really help you because it really depends on details. I am just showing you how complicated the things could be if you ask a really vague question.
