Sum[k=0..n, Sum[i>=0, C(2n,n-i(k+2))-C(2n,n+i(k+2)+k+1) ]].

Terms

a(0) =1a(1) =3a(2) =14a(3) =64a(4) =287a(5) =1265a(6) =5500a(7) =23660a(8) =100931a(9) =427669a(10) =1802170a(11) =7559473a(12) =31586889a(13) =131548937a(14) =546297862a(15) =2263042926a(16) =9354234285a(17) =38590884287a(18) =158932839678a(19) =653540335951