Eulerian matrix of size 6
[ 1 1 1 1 1 1]
[ 1 4 11 26 57 120]
[ 1 11 66 302 1191 4293]
[ 1 26 302 2416 15619 88234]
[ 1 57 1191 15619 156190 1310354]
[ 1 120 4293 88234 1310354 15724248]
Corresponding Eulerian triangle:
[ 0 0 0 0 0 0 0 0 0 0 0 0]
[ 1 0 0 0 0 0 0 0 0 0 0 0]
[ 1 1 0 0 0 0 0 0 0 0 0 0]
[ 1 4 1 0 0 0 0 0 0 0 0 0]
[ 1 11 11 1 0 0 0 0 0 0 0 0]
[ 1 26 66 26 1 0 0 0 0 0 0 0]
[ 1 57 302 302 57 1 0 0 0 0 0 0]
[ 1 120 1191 2416 1191 120 1 0 0 0 0 0]
[ 1 247 4293 15619 15619 4293 247 1 0 0 0 0]
[ 1 502 14608 88234 156190 88234 14608 502 1 0 0 0]
[ 1 1013 47840 455192 1310354 1310354 455192 47840 1013 1 0 0]
[ 1 2036 152637 2203488 9738114 15724248 9738114 2203488 152637 2036 1 0]
We may check that Eulerian matrix is TP2, therefore log[A_ij] is tropical TP. Does this help anyone?
Also Eulerian matrix is tropical TP2. Does not even use induction.
Euler triangle is not tropical TP2. see 11,1, 66 and 26 square.