Volume of the Birkhoff polytope for n =1, ..., 10

These are the top coefficients of the corresponding Ehrhart polynomials, multiplied by the unit volume nn-1.

n Vn
1 1
2 2
3 9 / 8
4 176 / 2835
5 23590375 / 167382319104
6 9700106723 / 1319281996032000000
7 77436678274508929033 / 13730296368223523839986892800000000
8 5562533838576105333259507434329 / 125890362600954779500814809426933398033089280000000000
9 559498129702796022246895686372766052475496691 / 215330276631180889478121101750832506606140689157723348094523801600000000000000
10 727291284016786420977508457990121862548823260052557333386607889 / 828160860106766855125676318796872729344622463533089422677980721388055739956270293750883504892820848640000000