A144054 Number of n X n binary arrays, symmetric under horizontal and vertical reflection, with every 1 adjacent to at least one other 1, but at most one 1 adjacent horizontally and at most one 1 adjacent vertically.

1, 2, 1, 5, 15, 65, 233, 1687, 8512, 96837, 990941, 16983353, 265775404, 7061955223, 183338740300, 7436406032400, 326859686075965, 20333610181788816, 1471382970077109810, 140578109521427255473, 16886941052098865528443