a(n) = n! * Sum_{i,j,k,l >= 0, i+j+k+l = n} 1/(i!*j!*k!*l!).

Terms

a(0) =1a(1) =1a(2) =3a(3) =10a(4) =47a(5) =126a(6) =522a(7) =1821a(8) =8143a(9) =26326a(10) =109958a(11) =396111a(12) =1737122a(13) =5998955a(14) =24949277a(15) =91979985a(16) =397402223a(17) =1418993350a(18) =5881338702a(19) =22010456331a(20) =94022106862a(21) =342803313261