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

Terms

a(0) =4a(1) =5a(2) =8a(3) =13a(4) =24a(5) =44a(6) =83a(7) =158a(8) =303a(9) =582a(10) =1120a(11) =2157a(12) =4156a(13) =8009a(14) =15436a(15) =29752a(16) =57347a(17) =110538a(18) =213067a(19) =410698a(20) =791644a(21) =1525941a(22) =2941344a(23) =5669621a(24) =10928544a(25) =21065444a(26) =40604947a(27) =78268550a(28) =150867479a(29) =290806414