Coefficient of x^n in g.f.^n is A000172(n).

Terms

a(0) =1a(1) =2a(2) =3a(3) =4a(4) =9a(5) =24a(6) =75a(7) =252a(8) =903a(9) =3376a(10) =13068a(11) =51960a(12) =211222a(13) =874440a(14) =3676335a(15) =15660680a(16) =67474980a(17) =293617248a(18) =1288876879a(19) =5701688928a(20) =25397905302a(21) =113838544880a(22) =513117505278