a(n) = Sum_{i=1..n} Sum_{j=1..n} (-1)^(i+j) * (i+j)!/(2!*i!*j!).

Terms

a(0) =0a(1) =1a(2) =1a(3) =5a(4) =15a(5) =56a(6) =203a(7) =757a(8) =2839a(9) =10736a(10) =40821a(11) =155948a(12) =598065a(13) =2301118a(14) =8878591a(15) =34340085a(16) =133100055a(17) =516851528a(18) =2010358061a(19) =7831136920a(20) =30546063745a(21) =119291436738a(22) =466379022561