How to use :not selector with :nth-of-type

How to use :not selector with :nth-of-type

Im about styling a table. I want each odd row to have a specific
background except the first row which contains headers.
I have used the following code which does not work:
.new-items-table tr:nth-of-type(odd):not(.new-items-table tr):nth-of-type(1)
{
background-color: red;
}