a(n) = Sum_{k=1..n} T(k, k-1), where T is the array in A026120.

Terms

a(0) =1a(1) =2a(2) =6a(3) =16a(4) =44a(5) =120a(6) =329a(7) =904a(8) =2493a(9) =6898a(10) =19151a(11) =53340a(12) =149019a(13) =417522a(14) =1172979a(15) =3303696a(16) =9326931a(17) =26390070a(18) =74824584a(19) =212565864a(20) =604972998a(21) =1724739940a(22) =4925066567a(23) =14085122376a(24) =40339596755a(25) =115688495806a(26) =332203673142a(27) =955090832416