Sum[i=0..n, Sum[j=0..n, C(n-i,i+j) * C(n-j,i+j) ]].

Terms

a(0) =1a(1) =1a(2) =5a(3) =14a(4) =46a(5) =145a(6) =462a(7) =1465a(8) =4641a(9) =14671a(10) =46298a(11) =145858a(12) =458811a(13) =1441204a(14) =4521255a(15) =14167145a(16) =44344445a(17) =138665650a(18) =433218100a(19) =1352333899a(20) =4218220602a(21) =13148256139a(22) =40956562683a(23) =127501720725a(24) =396702130246