Preheat the oven to 425 degrees. Grease a 9-inch-square pan generously. In a bowl whisk together the cornmeal, flour, baking powder, baking soda and the salt. In a small bowl whisk together the eggs, buttermilk and the butter, add the buttermilk mixture to the cornmeal mixture, and stir the batter until it is just combined. Heat the greased pan in the oven for 3 to 5 minutes, or until it is very hot, add the batter, spreading evenly and bake the cornbread in the middle of the oven for 15 minutes, or until the top is pale golden and the sides begin to pull away from the edges of the pan. Let the cornbread cool for 5 minutes, turn it out onto a rack and let it cool completely. Cut the cornbread into 9 squares.