a(n) = Sum_{k=0..2n-1} T(n,k)*T(n,k+1), T given by A027926.

Terms

a(0) =2a(1) =9a(2) =36a(3) =140a(4) =542a(5) =2100a(6) =8152a(7) =31707a(8) =123542a(9) =482119a(10) =1884060a(11) =7371715a(12) =28874778a(13) =113213129a(14) =444285136a(15) =1744931000a(16) =6858286418a(17) =26974115244a(18) =106157250796a(19) =418025117820