Binomial sum related to rooted trees.

Terms

a(0) =0a(1) =2a(2) =3a(3) =14a(4) =45a(5) =167a(6) =609a(7) =2270a(8) =8517a(9) =32207a(10) =122463a(11) =467843a(12) =1794195a(13) =6903353a(14) =26635773a(15) =103020254a(16) =399300165a(17) =1550554583a(18) =6031074183a(19) =23493410759a(20) =91638191235a(21) =357874310213