a(n) = Sum_{k=0..floor(n/4)} Stirling2(n,4*k).

Terms

a(0) =1a(1) =0a(2) =0a(3) =0a(4) =1a(5) =10a(6) =65a(7) =350a(8) =1702a(9) =7806a(10) =34855a(11) =157630a(12) =770529a(13) =4432220a(14) =31307432a(15) =259090260a(16) =2316320073a(17) =21172354778a(18) =193091210857