a(n) = Sum_{j=0..i, i=0..n} T(i,j), where T is the array in A026374.

Terms

a(0) =1a(1) =3a(2) =8a(3) =18a(4) =43a(5) =93a(6) =218a(7) =468a(8) =1093a(9) =2343a(10) =5468a(11) =11718a(12) =27343a(13) =58593a(14) =136718a(15) =292968a(16) =683593a(17) =1464843a(18) =3417968a(19) =7324218a(20) =17089843a(21) =36621093a(22) =85449218a(23) =183105468a(24) =427246093a(25) =915527343a(26) =2136230468a(27) =4577636718a(28) =10681152343a(29) =22888183593