Poincaré series [or Poincare series] for depths of roots in a certain root system.

Terms

a(0) =4a(1) =4a(2) =5a(3) =6a(4) =8a(5) =11a(6) =15a(7) =21a(8) =30a(9) =43a(10) =62a(11) =90a(12) =131a(13) =191a(14) =279a(15) =408a(16) =597a(17) =874a(18) =1280a(19) =1875a(20) =2747a(21) =4025a(22) =5898a(23) =8643a(24) =12666a(25) =18562a(26) =27203a(27) =39867a(28) =58427a(29) =85628