You need to check with an Oklahoma lawyer to get a completely accurate answer.
But in most states your husband's children would be entitled to a share of the estate (in California they could get as much as 2/3 of his property).
If the house is owned in "joint tenancy" (sometimes called "joint tenancy with rights of survivorship"), then the person who lives longest inherits the entire house.
But that's usually a lousy way to establish who-gets-what in a second marriage where there are "your" kids and "my" kids, and sometimes even "our" kids.
You and your husband need to consult with a knowledgeable lawyer immediately to protect your respective interests!!
This information is not intended to substitute for professional legal advice and does not create an attorney-client relationship. You should accept legal advice only from a licensed legal professional with whom you have an attorney-client relationship.
You need to check the state's intestate rules. In many states minor children or children from a previous marriage are protected and receive various assets. How that would apply to the home would be based upon the state the decedent lived in at the time of his or her death.
The best way to protect your rights and yours and your husband's wishes is to complete an appropriate estate plan with an attorney who specializes in this area. The rights of children from a previous marriage and spouses in the case where there is no will is state specific. Most states give substantial property rights to children where there is no will. Some states grant the surviving spouse the right to a life estate in the homestead, but again, the best plan is to make an estate plan, and follow through with all the instructions given to you by your attorney to execute the documents and complete any tasks necessary to complete your plan after execution of the planning documents.