a(n) = Sum_{k=0..floor(n/2)} T(n,k), T given by A026758.

Terms

a(0) =1a(1) =1a(2) =3a(3) =5a(4) =13a(5) =24a(6) =59a(7) =115a(8) =273a(9) =552a(10) =1278a(11) =2655a(12) =6031a(13) =12795a(14) =28632a(15) =61775a(16) =136572a(17) =298764a(18) =653948a(19) =1447225a(20) =3141427a(21) =7020833a(22) =15132512a(23) =34106865a(24) =73069892a(25) =165903082a(26) =353576829a(27) =807957495a(28) =1714132308a(29) =3939206346