a(n) = Sum[Sum[(i+j)!/i!/j!,{i,1,j}],{j,1,n}].

Terms

a(0) =2a(1) =11a(2) =45a(3) =170a(4) =631a(5) =2346a(6) =8780a(7) =33089a(8) =125466a(9) =478181a(10) =1830258a(11) =7030557a(12) =27088856a(13) =104647615a(14) =405187809a(15) =1571990918a(16) =6109558567a(17) =23782190466a(18) =92705454875a(19) =361834392094