a(n) = Sum_{k=0..floor(n/3)} binomial(n,k) * binomial(n+2*k,n-3*k).

Terms

a(0) =1a(1) =1a(2) =1a(3) =4a(4) =25a(5) =106a(6) =352a(7) =1114a(8) =3865a(9) =14539a(10) =54886a(11) =201763a(12) =732568a(13) =2679535a(14) =9917818a(15) =36903049a(16) =137265337a(17) =510201961a(18) =1898730307a(19) =7082472358a(20) =26468394430a(21) =99026247688a(22) =370771000975